maclaurin tan(θ)

{ "query": { "display": "maclaurin $$\\tan\\left(θ\\right)$$", "symbolab_question": "TAYLOR#maclaurin \\tan(θ)" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Taylor", "subTopic": "Maclaurin", "default": "θ+\\frac{1}{3}θ^{3}+\\frac{2}{15}θ^{5}+\\frac{17}{315}θ^{7}+\\frac{62}{2835}θ^{9}+\\ldots ", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "The Maclaurin Series of $$\\tan\\left(θ\\right):{\\quad}θ+\\frac{1}{3}θ^{3}+\\frac{2}{15}θ^{5}+\\frac{17}{315}θ^{7}+\\frac{62}{2835}θ^{9}+\\ldots\\:$$", "steps": [ { "type": "definition", "title": "Maclaurin Series", "text": "Taylor series of function f(x) at a is defined as:<br/>$$f\\left(x\\right)=f\\left(a\\right)+\\frac{f^{^{\\prime}}\\left(a\\right)}{1!}\\left(x-a\\right)+\\frac{f^{^{\\prime\\prime}}\\left(a\\right)}{2!}\\left(x-a\\right)^{2}+\\frac{f^{^{\\prime\\prime\\prime}}\\left(a\\right)}{3!}\\left(x-a\\right)^{3}+\\ldots$$<br/> <br/>Maclaurin series of function f(x) is a Taylor series of function f(x) at: a=0<br/>$$f\\left(x\\right)=f\\left(0\\right)+\\frac{f^{^{\\prime}}\\left(0\\right)}{1!}\\left(x\\right)+\\frac{f^{^{\\prime\\prime}}\\left(0\\right)}{2!}\\left(x\\right)^{2}+\\frac{f^{^{\\prime\\prime\\prime}}\\left(0\\right)}{3!}\\left(x\\right)^{3}+\\ldots$$" }, { "type": "interim", "title": "Apply the Maclaurin Formula", "steps": [ { "type": "step", "primary": "Find the derivatives of $$f\\left(x\\right)=\\tan\\left(θ\\right),\\:$$at $$a=0$$" }, { "type": "interim", "title": "$$f\\left(0\\right):{\\quad}0$$", "steps": [ { "type": "step", "primary": "Take the point $$θ=0$$ and plug it into $$\\tan\\left(θ\\right)$$", "result": "=\\tan\\left(0\\right)" }, { "type": "step", "primary": "Use the following trivial identity:$${\\quad}\\tan\\left(0\\right)=0$$", "secondary": [ "$$\\tan\\left(x\\right)$$ periodicity table with $$πn$$ cycle:<br/>$$\\begin{array}{|c|c|}\\hline x&\\tan(x)\\\\\\hline 0&0\\\\\\hline \\frac{π}{6}&\\frac{\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&1\\\\\\hline \\frac{π}{3}&\\sqrt{3}\\\\\\hline \\frac{π}{2}&\\pm\\infty\\\\\\hline \\frac{2π}{3}&-\\sqrt{3}\\\\\\hline \\frac{3π}{4}&-1\\\\\\hline \\frac{5π}{6}&-\\frac{\\sqrt{3}}{3}\\\\\\hline &\\\\\\hline \\end{array}$$" ], "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver" } }, { "type": "step", "result": "=0+\\frac{\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)\\left(0\\right)}{1!}θ+\\frac{\\frac{d^{2}}{dθ^{2}}\\left(\\tan\\left(θ\\right)\\right)\\left(0\\right)}{2!}θ^{2}+\\frac{\\frac{d^{3}}{dθ^{3}}\\left(\\tan\\left(θ\\right)\\right)\\left(0\\right)}{3!}θ^{3}+\\ldots\\:" } ], "meta": { "interimType": "Maclaurin Apply Formula 0Eq" } }, { "type": "step", "result": "=0+\\frac{\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)\\left(0\\right)}{1!}θ+\\frac{\\frac{d^{2}}{dθ^{2}}\\left(\\tan\\left(θ\\right)\\right)\\left(0\\right)}{2!}θ^{2}+\\frac{\\frac{d^{3}}{dθ^{3}}\\left(\\tan\\left(θ\\right)\\right)\\left(0\\right)}{3!}θ^{3}+\\ldots\\:" }, { "type": "interim", "title": "Evaluate Derivatives", "steps": [ { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)\\left(0\\right){\\quad:\\quad}1$$", "input": "\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)\\left(0\\right)", "steps": [ { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "result": "=\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYs0Vsx3c0nt53Utu5DgHTe3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK00HV/bLwwglXz/2YNx2i4JHggSrmCoGjSdud4CyOvVOgdQN0k+N3NHNAkEIoLrhB631k80nn+ODwI4ID0kCF5+z7zRz81SqRD02HEUE4Wqh" } }, { "type": "step", "result": "=\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "Evaluate $$\\sec^{2}\\left(θ\\right)\\:$$at point $$θ=0:{\\quad}1$$", "result": "=1", "steps": [ { "type": "step", "primary": "Take the point $$θ=0$$ and plug it into $$\\sec^{2}\\left(θ\\right)$$", "result": "=\\sec^{2}\\left(0\\right)" }, { "type": "interim", "title": "Simplify $$\\sec\\left(0\\right):{\\quad}1$$", "input": "\\sec\\left(0\\right)", "result": "=1^{2}", "steps": [ { "type": "step", "primary": "Use the following trivial identity:$${\\quad}\\sec\\left(0\\right)=1$$", "secondary": [ "$$\\sec\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sec(x)&x&\\sec(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{2\\sqrt{3}}{3}&\\frac{7π}{6}&-\\frac{2\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&\\sqrt{2}&\\frac{5π}{4}&-\\sqrt{2}\\\\\\hline \\frac{π}{3}&2&\\frac{4π}{3}&-2\\\\\\hline \\frac{π}{2}&\\mathrm{Undefined}&\\frac{3π}{2}&\\mathrm{Undefined}\\\\\\hline \\frac{2π}{3}&-2&\\frac{5π}{3}&2\\\\\\hline \\frac{3π}{4}&-\\sqrt{2}&\\frac{7π}{4}&\\sqrt{2}\\\\\\hline \\frac{5π}{6}&-\\frac{2\\sqrt{3}}{3}&\\frac{11π}{6}&\\frac{2\\sqrt{3}}{3}\\\\\\hline \\end{array}$$" ], "result": "=1" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Laplace Eval Function At Point Specific 3Eq" } }, { "type": "step", "result": "1" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\frac{d^{2}}{dθ^{2}}\\left(\\tan\\left(θ\\right)\\right)\\left(0\\right){\\quad:\\quad}0$$", "input": "\\frac{d^{2}}{dθ^{2}}\\left(\\tan\\left(θ\\right)\\right)\\left(0\\right)", "steps": [ { "type": "interim", "title": "$$\\frac{d^{2}}{dθ^{2}}\\left(\\tan\\left(θ\\right)\\right)=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d^{2}}{dθ^{2}}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "result": "=\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYs0Vsx3c0nt53Utu5DgHTe3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK00HV/bLwwglXz/2YNx2i4JHggSrmCoGjSdud4CyOvVOgdQN0k+N3NHNAkEIoLrhB631k80nn+ODwI4ID0kCF5+z7zRz81SqRD02HEUE4Wqh" } }, { "type": "step", "result": "=\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{2},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{2}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{2}\\right)=2u$$", "input": "\\frac{d}{du}\\left(u^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2u^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2u", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYr+VZUwBnLdzbS6DZQ+f4s+k3hxk9aCfAWodBRxXgUexMchyqTAJWrzJaDbnNcFsJUeCBKuYKgaNJ253gLI69U79qbCA2QqVmvm3jGRXZ2ppvbGT4j1utMEkCDH25m/vlQ==" } }, { "type": "step", "result": "=2u\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgCSXIThcWH58oYoMPt18lzFqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidZClirlWNFPnHR1rvdyoFLI7IC+oqnwb1T6nPLpiw1eysw2NXPd8jZnr3fcPpqc7ejeh7+jKEzLb7VNCEMF3Z/Maswa5oIagKQFOfrnKm5+Gbb/PxX3awut5KxRxyZf8A=" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{1+1}\\left(θ\\right)$$" ], "result": "=2\\sec^{1+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7aY4rYekMZhUfP6sZ6mfqd6Lsz0lt1XLEr88Kc/cjJo5V00rpv8+ZC6TM10tVCSHspqrCcTwb/I0f8OwMjuU6eBAUa+AHsHd5OGpMiuFITjOuqvW3hAHH635V/vEEYIatHimBRYRqHSWeJkuUPhfTC5LQ0B1JNgsb5SXzbS+SCnAfGB9H25Zc+Ua1z7Kw8kgb4gBJl4WMO1rA0a30/bUYlg==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "Evaluate $$2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\:$$at point $$θ=0:{\\quad}0$$", "result": "=0", "steps": [ { "type": "step", "primary": "Take the point $$θ=0$$ and plug it into $$2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=2\\sec^{2}\\left(0\\right)\\tan\\left(0\\right)" }, { "type": "interim", "title": "$$\\sec^{2}\\left(0\\right)=1$$", "input": "\\sec^{2}\\left(0\\right)", "steps": [ { "type": "interim", "title": "Simplify $$\\sec\\left(0\\right):{\\quad}1$$", "input": "\\sec\\left(0\\right)", "result": "=1^{2}", "steps": [ { "type": "step", "primary": "Use the following trivial identity:$${\\quad}\\sec\\left(0\\right)=1$$", "secondary": [ "$$\\sec\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sec(x)&x&\\sec(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{2\\sqrt{3}}{3}&\\frac{7π}{6}&-\\frac{2\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&\\sqrt{2}&\\frac{5π}{4}&-\\sqrt{2}\\\\\\hline \\frac{π}{3}&2&\\frac{4π}{3}&-2\\\\\\hline \\frac{π}{2}&\\mathrm{Undefined}&\\frac{3π}{2}&\\mathrm{Undefined}\\\\\\hline \\frac{2π}{3}&-2&\\frac{5π}{3}&2\\\\\\hline \\frac{3π}{4}&-\\sqrt{2}&\\frac{7π}{4}&\\sqrt{2}\\\\\\hline \\frac{5π}{6}&-\\frac{2\\sqrt{3}}{3}&\\frac{11π}{6}&\\frac{2\\sqrt{3}}{3}\\\\\\hline \\end{array}$$" ], "result": "=1" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7y1xaZeQTSzp609voOcYk6VXTSum/z5kLpMzXS1UJIexiYFOaxxQqg3o8CNKkTm2qW+Sc57RV820557NJqaC71BfVjTo90KMWspnEdE4CKjE=" } }, { "type": "step", "result": "=2\\cdot\\:1\\cdot\\:\\tan\\left(0\\right)" }, { "type": "interim", "title": "Simplify $$\\tan\\left(0\\right):{\\quad}0$$", "input": "\\tan\\left(0\\right)", "result": "=2\\cdot\\:1\\cdot\\:0", "steps": [ { "type": "step", "primary": "Use the following trivial identity:$${\\quad}\\tan\\left(0\\right)=0$$", "secondary": [ "$$\\tan\\left(x\\right)$$ periodicity table with $$πn$$ cycle:<br/>$$\\begin{array}{|c|c|}\\hline x&\\tan(x)\\\\\\hline 0&0\\\\\\hline \\frac{π}{6}&\\frac{\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&1\\\\\\hline \\frac{π}{3}&\\sqrt{3}\\\\\\hline \\frac{π}{2}&\\pm\\infty\\\\\\hline \\frac{2π}{3}&-\\sqrt{3}\\\\\\hline \\frac{3π}{4}&-1\\\\\\hline \\frac{5π}{6}&-\\frac{\\sqrt{3}}{3}\\\\\\hline &\\\\\\hline \\end{array}$$" ], "result": "=0" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Apply rule $$0\\cdot\\:a=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Laplace Eval Function At Point Specific 3Eq" } }, { "type": "step", "result": "0" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\frac{d^{3}}{dθ^{3}}\\left(\\tan\\left(θ\\right)\\right)\\left(0\\right){\\quad:\\quad}2$$", "input": "\\frac{d^{3}}{dθ^{3}}\\left(\\tan\\left(θ\\right)\\right)\\left(0\\right)", "steps": [ { "type": "interim", "title": "$$\\frac{d^{3}}{dθ^{3}}\\left(\\tan\\left(θ\\right)\\right)=-4\\sec^{2}\\left(θ\\right)+6\\sec^{4}\\left(θ\\right)$$", "input": "\\frac{d^{3}}{dθ^{3}}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "result": "=\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYs0Vsx3c0nt53Utu5DgHTe3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK00HV/bLwwglXz/2YNx2i4JHggSrmCoGjSdud4CyOvVOgdQN0k+N3NHNAkEIoLrhB631k80nn+ODwI4ID0kCF5+z7zRz81SqRD02HEUE4Wqh" } }, { "type": "step", "result": "=\\frac{d^{2}}{dθ^{2}}\\left(\\sec^{2}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{2},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{2}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{2}\\right)=2u$$", "input": "\\frac{d}{du}\\left(u^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2u^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2u", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYr+VZUwBnLdzbS6DZQ+f4s+k3hxk9aCfAWodBRxXgUexMchyqTAJWrzJaDbnNcFsJUeCBKuYKgaNJ253gLI69U79qbCA2QqVmvm3jGRXZ2ppvbGT4j1utMEkCDH25m/vlQ==" } }, { "type": "step", "result": "=2u\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgCSXIThcWH58oYoMPt18lzFqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidZClirlWNFPnHR1rvdyoFLI7IC+oqnwb1T6nPLpiw1eysw2NXPd8jZnr3fcPpqc7ejeh7+jKEzLb7VNCEMF3Z/Maswa5oIagKQFOfrnKm5+Gbb/PxX3awut5KxRxyZf8A=" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{1+1}\\left(θ\\right)$$" ], "result": "=2\\sec^{1+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7aY4rYekMZhUfP6sZ6mfqd6Lsz0lt1XLEr88Kc/cjJo5V00rpv8+ZC6TM10tVCSHspqrCcTwb/I0f8OwMjuU6eBAUa+AHsHd5OGpMiuFITjOuqvW3hAHH635V/vEEYIatHimBRYRqHSWeJkuUPhfTC5LQ0B1JNgsb5SXzbS+SCnAfGB9H25Zc+Ua1z7Kw8kgb4gBJl4WMO1rA0a30/bUYlg==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{d}{dθ}\\left(2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\right)=-4\\sec^{2}\\left(θ\\right)+6\\sec^{4}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=2\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the Product Rule: $$\\left(f{\\cdot}g\\right)'=f'{\\cdot}g+f{\\cdot}g'$$", "secondary": [ "$$f=\\sec^{2}\\left(θ\\right),\\:g=\\tan\\left(θ\\right)$$" ], "result": "=2\\left(\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)\\tan\\left(θ\\right)+\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)\\sec^{2}\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Product%20Rule", "practiceTopic": "Product Rule" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{2},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{2}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{2}\\right)=2u$$", "input": "\\frac{d}{du}\\left(u^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2u^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2u", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYr+VZUwBnLdzbS6DZQ+f4s+k3hxk9aCfAWodBRxXgUexMchyqTAJWrzJaDbnNcFsJUeCBKuYKgaNJ253gLI69U79qbCA2QqVmvm3jGRXZ2ppvbGT4j1utMEkCDH25m/vlQ==" } }, { "type": "step", "result": "=2u\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgCSXIThcWH58oYoMPt18lzFqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidZClirlWNFPnHR1rvdyoFLI7IC+oqnwb1T6nPLpiw1eysw2NXPd8jZnr3fcPpqc7ejeh7+jKEzLb7VNCEMF3Z/Maswa5oIagKQFOfrnKm5+Gbb/PxX3awut5KxRxyZf8A=" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{1+1}\\left(θ\\right)$$" ], "result": "=2\\sec^{1+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7aY4rYekMZhUfP6sZ6mfqd6Lsz0lt1XLEr88Kc/cjJo5V00rpv8+ZC6TM10tVCSHspqrCcTwb/I0f8OwMjuU6eBAUa+AHsHd5OGpMiuFITjOuqvW3hAHH635V/vEEYIatHimBRYRqHSWeJkuUPhfTC5LQ0B1JNgsb5SXzbS+SCnAfGB9H25Zc+Ua1z7Kw8kgb4gBJl4WMO1rA0a30/bUYlg==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "result": "=\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYs0Vsx3c0nt53Utu5DgHTe3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK00HV/bLwwglXz/2YNx2i4JHggSrmCoGjSdud4CyOvVOgdQN0k+N3NHNAkEIoLrhB631k80nn+ODwI4ID0kCF5+z7zRz81SqRD02HEUE4Wqh" } }, { "type": "step", "result": "=2\\left(2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Simplify $$2\\left(2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\right):{\\quad}-4\\sec^{2}\\left(θ\\right)+6\\sec^{4}\\left(θ\\right)$$", "input": "2\\left(2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\right)", "result": "=-4\\sec^{2}\\left(θ\\right)+6\\sec^{4}\\left(θ\\right)", "steps": [ { "type": "interim", "title": "$$2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)=2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)$$", "input": "2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\tan\\left(θ\\right)\\tan\\left(θ\\right)=\\:\\tan^{1+1}\\left(θ\\right)$$" ], "result": "=2\\sec^{2}\\left(θ\\right)\\tan^{1+1}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Md1SJKHYLxhOYRRbsLTu+axUZgt6FTmFSHnRsotkqemLeJKmeakO1+Q3aSIldG41s7BwSVdKEYSqn+4fKAQoGk0xmFIl1KTO5aAkhaHvKzBALhShbMGZIZsZWTO721HsWI4dG63F/aSz/PFDvu5FnO3sngYNzcydIedGQjjtM8sBMUhkLMqDD6KHvgZAgKMdRQzIIgXxnWeDhDTk2Lfum8RPUSGriaFRn0NZXHZrPQw=" } }, { "type": "interim", "title": "$$\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)=\\sec^{4}\\left(θ\\right)$$", "input": "\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)=\\:\\sec^{2+2}\\left(θ\\right)$$" ], "result": "=\\sec^{2+2}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+2=4$$", "result": "=\\sec^{4}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7AtrsReczPdUqIdn5/Ev5mFtNfVJqszzpMwg6spHO8F8gJ/ZZA32ZInFBpDtxBfiKn3Jyj2VJgBojKQHjINpkkETYvnjqwhFu6aWmDOYpAVMO/U6qyRZonn1WPQ/21pz3cKXVLLQXMot2gtnZH7qY5zkbLioo7YVO0SRnDAO/7Uokt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "=2\\left(\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Expand $$\\left(\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)\\cdot\\:2:{\\quad}2\\sec^{4}\\left(θ\\right)+4\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)$$", "input": "\\left(\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)\\cdot\\:2", "result": "=2\\sec^{4}\\left(θ\\right)+4\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)", "steps": [ { "type": "step", "result": "=2\\left(\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the distributive law: $$a\\left(b+c\\right)=ab+ac$$", "secondary": [ "$$a=2,\\:b=\\sec^{4}\\left(θ\\right),\\:c=2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)$$" ], "result": "=2\\sec^{4}\\left(θ\\right)+2\\cdot\\:2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=2\\sec^{4}\\left(θ\\right)+4\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7cTiUO1fOFax2wXHZewjhxEUAdh/O2PLAxzMjSpKce9YL4tWjzbYCZhBh/Hc2rUXacMKakGXZAry+N/hvJOodjXCQoYlYQ8U+Tfyx0kyzI8iF3F2TkMs7mKjIsUPAR/Vd82dH/qvnCC7/oZy4HZlX2aHh6acbPp8G70jj/skoFx7vbBmbuQNTF0TphKZ8Ruvacu+LCkBvFQsw7xBL2Ys2TUYb1899n2cPVSJTSzzdZzp+38s+uQz1W6FLDk3KkIYWhJuRFkG6Ai089lDjeB1lMbCI2sSeA74029n2yo277ZU=" } }, { "type": "interim", "title": "Rewrite using trig identities", "input": "2\\sec^{4}\\left(θ\\right)+4\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)", "result": "=-4\\sec^{2}\\left(θ\\right)+6\\sec^{4}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Use the Pythagorean identity: $$\\tan^{2}\\left(x\\right)+1=\\sec^{2}\\left(x\\right)$$", "secondary": [ "$$\\tan^{2}\\left(x\\right)=\\sec^{2}\\left(x\\right)-1$$" ], "result": "=2\\sec^{4}\\left(θ\\right)+4\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right)" }, { "type": "interim", "title": "Simplify $$2\\sec^{4}\\left(θ\\right)+4\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right):{\\quad}6\\sec^{4}\\left(θ\\right)-4\\sec^{2}\\left(θ\\right)$$", "input": "2\\sec^{4}\\left(θ\\right)+4\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right)", "result": "=6\\sec^{4}\\left(θ\\right)-4\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "interim", "title": "Expand $$4\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right):{\\quad}4\\sec^{4}\\left(θ\\right)-4\\sec^{2}\\left(θ\\right)$$", "input": "4\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right)", "result": "=2\\sec^{4}\\left(θ\\right)+4\\sec^{4}\\left(θ\\right)-4\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b-c\\right)=ab-ac$$", "secondary": [ "$$a=4\\sec^{2}\\left(θ\\right),\\:b=\\sec^{2}\\left(θ\\right),\\:c=1$$" ], "result": "=4\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)-4\\sec^{2}\\left(θ\\right)\\cdot\\:1", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "result": "=4\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)-4\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$4\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)-4\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right):{\\quad}4\\sec^{4}\\left(θ\\right)-4\\sec^{2}\\left(θ\\right)$$", "input": "4\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)-4\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right)", "result": "=4\\sec^{4}\\left(θ\\right)-4\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "interim", "title": "$$4\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)=4\\sec^{4}\\left(θ\\right)$$", "input": "4\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)=\\:\\sec^{2+2}\\left(θ\\right)$$" ], "result": "=4\\sec^{2+2}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+2=4$$", "result": "=4\\sec^{4}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vjDr4p6TvcgiSLV1uDo4U9JYPDPDesonRI520G8zYRPehkKrn0era9rz8TlL+x/vMHVZf8xiryCPkbUIYXhiIHRfFgXB+vlRb/AHCJklpF9e7YF3TFAJSjcLX2VvhbubqX/cSuhK5Ty2xqd/IdNr6zlZ8CVoKZl2BAoDdAtcFNxU2fg2Eo8khoQfnK/P3ock" } }, { "type": "interim", "title": "$$4\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right)=4\\sec^{2}\\left(θ\\right)$$", "input": "4\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:1=4$$", "result": "=4\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7xF9rlQkuAOrRzKTf6eQs7fgkIqVBCzsTrV5i/ECNCw8tOtZYwUjyXhDTsNnn6Elr/BCsllMMEyc8UGKoUwuYxu6vfsn/ogOG+fHMfNQh9JeZfoRPsbCjih+xtMTOMGl4tzgLeG7CLgj8aHpc44iME/LWpEthEIQrAjNHY4qDKt/ET1Ehq4mhUZ9DWVx2az0M" } }, { "type": "step", "result": "=4\\sec^{4}\\left(θ\\right)-4\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vjDr4p6TvcgiSLV1uDo4U/jbMbEoNNbIR/7Q84OynCfTLx8mOdHYVzxX643JqKFIvTZkYD4gUeSoFrhGGhxq98kwYX4VRPb1geKkn+NU7mcE3SiHEhDlxIXJDVY+PPdvgQUxJPyUNnGfVirkcwpVO84fwxlvni/T8DIWnD0mw9U6FFV/0Hm6Y1hI7VaPzzKhkSoZP7CyArJlOo0FRny79Q==" } }, { "type": "step", "primary": "Add similar elements: $$2\\sec^{4}\\left(θ\\right)+4\\sec^{4}\\left(θ\\right)=6\\sec^{4}\\left(θ\\right)$$", "result": "=6\\sec^{4}\\left(θ\\right)-4\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7f48emNsIl4MI4hQZuZje4penFTHzh6X+DX3uvPdqYy7umM/GfBKdAtyuCOAeIKZmAJYpRu9XpYrd8NSAW2DdD0c/CXknyRKFbCevkP3vkuVyZXAw9UlCE0mEZtdsFPrV7q9+yf+iA4b58cx81CH0l/C30sSftAIFS6Qkpy19Ikp0x8irWzKvvppKMUvqTrnrUObZozYPTEmgGDR5V3grHDoUVX/QebpjWEjtVo/PMqGRKhk/sLICsmU6jQVGfLv1" } } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7xqDNX42OyJjPevbF+7IBng7708BkbVE1gb8/0kMEgzvXUIQv0aG2ncbDgSWX/DWycNZx/2EDub2dSsLw3NGg9xLC2T8NFZvTA+pff5AdXPdX9lRN4mlYq/zI0dgxEXGdPvdnhi0s3lZ+DJ7rEiTeqenxKTmAB1ahPFE542Dd/TeutOLi0kB5xHysaBsHIgntIaTy8gmKy5PY3el+RbUmlkmFP65WPUHCIGTpzT/EAccetg60+ZUlqP5jB+ReldUA5AMRxXWt+t9va0qZyagxyQ==" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7xI7dknI212svHgZHHIcoRvkdSragnZ5K8+WMRM5qjJ6ETkUamdUz59t0aHpV3uN0rE0Ib0rKQ2oM97vl/gDbowCWKUbvV6WK3fDUgFtg3Q+MfZ1BFRhhGnQf9q28g9nK/SzNF20Xrftd883pI0RALQ5+VkRmueXOP55Piqx5MyWBBTEk/JQ2cZ9WKuRzClU7m4RppR4RRNJR6+bqv/lHExUnakC9Y1jhr4uI97E9yvalDk6K3L00/zgNSSliALTtnpy3tAot7+5PIP9fBKZw2Llxun299aFzbjCyRFblUKE=" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=-4\\sec^{2}\\left(θ\\right)+6\\sec^{4}\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "Evaluate $$-4\\sec^{2}\\left(θ\\right)+6\\sec^{4}\\left(θ\\right)\\:$$at point $$θ=0:{\\quad}2$$", "result": "=2", "steps": [ { "type": "step", "primary": "Take the point $$θ=0$$ and plug it into $$-4\\sec^{2}\\left(θ\\right)+6\\sec^{4}\\left(θ\\right)$$", "result": "=-4\\sec^{2}\\left(0\\right)+6\\sec^{4}\\left(0\\right)" }, { "type": "interim", "title": "$$4\\sec^{2}\\left(0\\right)=4$$", "input": "4\\sec^{2}\\left(0\\right)", "steps": [ { "type": "interim", "title": "$$\\sec^{2}\\left(0\\right)=1$$", "input": "\\sec^{2}\\left(0\\right)", "steps": [ { "type": "interim", "title": "Simplify $$\\sec\\left(0\\right):{\\quad}1$$", "input": "\\sec\\left(0\\right)", "result": "=1^{2}", "steps": [ { "type": "step", "primary": "Use the following trivial identity:$${\\quad}\\sec\\left(0\\right)=1$$", "secondary": [ "$$\\sec\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sec(x)&x&\\sec(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{2\\sqrt{3}}{3}&\\frac{7π}{6}&-\\frac{2\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&\\sqrt{2}&\\frac{5π}{4}&-\\sqrt{2}\\\\\\hline \\frac{π}{3}&2&\\frac{4π}{3}&-2\\\\\\hline \\frac{π}{2}&\\mathrm{Undefined}&\\frac{3π}{2}&\\mathrm{Undefined}\\\\\\hline \\frac{2π}{3}&-2&\\frac{5π}{3}&2\\\\\\hline \\frac{3π}{4}&-\\sqrt{2}&\\frac{7π}{4}&\\sqrt{2}\\\\\\hline \\frac{5π}{6}&-\\frac{2\\sqrt{3}}{3}&\\frac{11π}{6}&\\frac{2\\sqrt{3}}{3}\\\\\\hline \\end{array}$$" ], "result": "=1" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7y1xaZeQTSzp609voOcYk6VXTSum/z5kLpMzXS1UJIexiYFOaxxQqg3o8CNKkTm2qW+Sc57RV820557NJqaC71BfVjTo90KMWspnEdE4CKjE=" } }, { "type": "step", "result": "=4\\cdot\\:1" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:1=4$$", "result": "=4" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ox/EiYQPVJOzyGidGjKBOy061ljBSPJeENOw2efoSWuqze2Yzn0Gom7AYUa+PIjkqeQiXiG14PkDGzHUEXsJxXLa+/gTW4OMPy481fO1c1k=" } }, { "type": "interim", "title": "$$6\\sec^{4}\\left(0\\right)=6$$", "input": "6\\sec^{4}\\left(0\\right)", "steps": [ { "type": "interim", "title": "$$\\sec^{4}\\left(0\\right)=1$$", "input": "\\sec^{4}\\left(0\\right)", "steps": [ { "type": "interim", "title": "Simplify $$\\sec\\left(0\\right):{\\quad}1$$", "input": "\\sec\\left(0\\right)", "result": "=1^{4}", "steps": [ { "type": "step", "primary": "Use the following trivial identity:$${\\quad}\\sec\\left(0\\right)=1$$", "secondary": [ "$$\\sec\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sec(x)&x&\\sec(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{2\\sqrt{3}}{3}&\\frac{7π}{6}&-\\frac{2\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&\\sqrt{2}&\\frac{5π}{4}&-\\sqrt{2}\\\\\\hline \\frac{π}{3}&2&\\frac{4π}{3}&-2\\\\\\hline \\frac{π}{2}&\\mathrm{Undefined}&\\frac{3π}{2}&\\mathrm{Undefined}\\\\\\hline \\frac{2π}{3}&-2&\\frac{5π}{3}&2\\\\\\hline \\frac{3π}{4}&-\\sqrt{2}&\\frac{7π}{4}&\\sqrt{2}\\\\\\hline \\frac{5π}{6}&-\\frac{2\\sqrt{3}}{3}&\\frac{11π}{6}&\\frac{2\\sqrt{3}}{3}\\\\\\hline \\end{array}$$" ], "result": "=1" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7TnbPBMsEd78kmmwfYDHqNVXTSum/z5kLpMzXS1UJIexiYFOaxxQqg3o8CNKkTm2qW+Sc57RV820557NJqaC71Ka6Ma3rg33eDIhfX/iuVnM=" } }, { "type": "step", "result": "=6\\cdot\\:1" }, { "type": "step", "primary": "Multiply the numbers: $$6\\cdot\\:1=6$$", "result": "=6" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s783g9I9kp1nOnUl2IcPYnsS061ljBSPJeENOw2efoSWtIfVhXcrdDNEQpHnsPIiJiDGx695N0IVbGe/5emi+wc1YkF2IscJkqTizeIl6vkBI=" } }, { "type": "step", "result": "=-4+6" }, { "type": "step", "primary": "Simplify" }, { "type": "step", "result": "=2" } ], "meta": { "solvingClass": "Solver", "interimType": "Laplace Eval Function At Point Specific 3Eq" } }, { "type": "step", "result": "2" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\frac{d^{4}}{dθ^{4}}\\left(\\tan\\left(θ\\right)\\right)\\left(0\\right){\\quad:\\quad}0$$", "input": "\\frac{d^{4}}{dθ^{4}}\\left(\\tan\\left(θ\\right)\\right)\\left(0\\right)", "steps": [ { "type": "interim", "title": "$$\\frac{d^{4}}{dθ^{4}}\\left(\\tan\\left(θ\\right)\\right)=-8\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+24\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d^{4}}{dθ^{4}}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "result": "=\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYs0Vsx3c0nt53Utu5DgHTe3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK00HV/bLwwglXz/2YNx2i4JHggSrmCoGjSdud4CyOvVOgdQN0k+N3NHNAkEIoLrhB631k80nn+ODwI4ID0kCF5+z7zRz81SqRD02HEUE4Wqh" } }, { "type": "step", "result": "=\\frac{d^{3}}{dθ^{3}}\\left(\\sec^{2}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{2},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{2}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{2}\\right)=2u$$", "input": "\\frac{d}{du}\\left(u^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2u^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2u", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYr+VZUwBnLdzbS6DZQ+f4s+k3hxk9aCfAWodBRxXgUexMchyqTAJWrzJaDbnNcFsJUeCBKuYKgaNJ253gLI69U79qbCA2QqVmvm3jGRXZ2ppvbGT4j1utMEkCDH25m/vlQ==" } }, { "type": "step", "result": "=2u\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgCSXIThcWH58oYoMPt18lzFqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidZClirlWNFPnHR1rvdyoFLI7IC+oqnwb1T6nPLpiw1eysw2NXPd8jZnr3fcPpqc7ejeh7+jKEzLb7VNCEMF3Z/Maswa5oIagKQFOfrnKm5+Gbb/PxX3awut5KxRxyZf8A=" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{1+1}\\left(θ\\right)$$" ], "result": "=2\\sec^{1+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7aY4rYekMZhUfP6sZ6mfqd6Lsz0lt1XLEr88Kc/cjJo5V00rpv8+ZC6TM10tVCSHspqrCcTwb/I0f8OwMjuU6eBAUa+AHsHd5OGpMiuFITjOuqvW3hAHH635V/vEEYIatHimBRYRqHSWeJkuUPhfTC5LQ0B1JNgsb5SXzbS+SCnAfGB9H25Zc+Ua1z7Kw8kgb4gBJl4WMO1rA0a30/bUYlg==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{d^{2}}{dθ^{2}}\\left(2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\right)=-4\\sec^{2}\\left(θ\\right)+6\\sec^{4}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=2\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the Product Rule: $$\\left(f{\\cdot}g\\right)'=f'{\\cdot}g+f{\\cdot}g'$$", "secondary": [ "$$f=\\sec^{2}\\left(θ\\right),\\:g=\\tan\\left(θ\\right)$$" ], "result": "=2\\left(\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)\\tan\\left(θ\\right)+\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)\\sec^{2}\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Product%20Rule", "practiceTopic": "Product Rule" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{2},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{2}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{2}\\right)=2u$$", "input": "\\frac{d}{du}\\left(u^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2u^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2u", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYr+VZUwBnLdzbS6DZQ+f4s+k3hxk9aCfAWodBRxXgUexMchyqTAJWrzJaDbnNcFsJUeCBKuYKgaNJ253gLI69U79qbCA2QqVmvm3jGRXZ2ppvbGT4j1utMEkCDH25m/vlQ==" } }, { "type": "step", "result": "=2u\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgCSXIThcWH58oYoMPt18lzFqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidZClirlWNFPnHR1rvdyoFLI7IC+oqnwb1T6nPLpiw1eysw2NXPd8jZnr3fcPpqc7ejeh7+jKEzLb7VNCEMF3Z/Maswa5oIagKQFOfrnKm5+Gbb/PxX3awut5KxRxyZf8A=" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{1+1}\\left(θ\\right)$$" ], "result": "=2\\sec^{1+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7aY4rYekMZhUfP6sZ6mfqd6Lsz0lt1XLEr88Kc/cjJo5V00rpv8+ZC6TM10tVCSHspqrCcTwb/I0f8OwMjuU6eBAUa+AHsHd5OGpMiuFITjOuqvW3hAHH635V/vEEYIatHimBRYRqHSWeJkuUPhfTC5LQ0B1JNgsb5SXzbS+SCnAfGB9H25Zc+Ua1z7Kw8kgb4gBJl4WMO1rA0a30/bUYlg==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "result": "=\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYs0Vsx3c0nt53Utu5DgHTe3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK00HV/bLwwglXz/2YNx2i4JHggSrmCoGjSdud4CyOvVOgdQN0k+N3NHNAkEIoLrhB631k80nn+ODwI4ID0kCF5+z7zRz81SqRD02HEUE4Wqh" } }, { "type": "step", "result": "=2\\left(2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Simplify $$2\\left(2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\right):{\\quad}-4\\sec^{2}\\left(θ\\right)+6\\sec^{4}\\left(θ\\right)$$", "input": "2\\left(2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\right)", "result": "=-4\\sec^{2}\\left(θ\\right)+6\\sec^{4}\\left(θ\\right)", "steps": [ { "type": "interim", "title": "$$2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)=2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)$$", "input": "2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\tan\\left(θ\\right)\\tan\\left(θ\\right)=\\:\\tan^{1+1}\\left(θ\\right)$$" ], "result": "=2\\sec^{2}\\left(θ\\right)\\tan^{1+1}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Md1SJKHYLxhOYRRbsLTu+axUZgt6FTmFSHnRsotkqemLeJKmeakO1+Q3aSIldG41s7BwSVdKEYSqn+4fKAQoGk0xmFIl1KTO5aAkhaHvKzBALhShbMGZIZsZWTO721HsWI4dG63F/aSz/PFDvu5FnO3sngYNzcydIedGQjjtM8sBMUhkLMqDD6KHvgZAgKMdRQzIIgXxnWeDhDTk2Lfum8RPUSGriaFRn0NZXHZrPQw=" } }, { "type": "interim", "title": "$$\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)=\\sec^{4}\\left(θ\\right)$$", "input": "\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)=\\:\\sec^{2+2}\\left(θ\\right)$$" ], "result": "=\\sec^{2+2}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+2=4$$", "result": "=\\sec^{4}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7AtrsReczPdUqIdn5/Ev5mFtNfVJqszzpMwg6spHO8F8gJ/ZZA32ZInFBpDtxBfiKn3Jyj2VJgBojKQHjINpkkETYvnjqwhFu6aWmDOYpAVMO/U6qyRZonn1WPQ/21pz3cKXVLLQXMot2gtnZH7qY5zkbLioo7YVO0SRnDAO/7Uokt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "=2\\left(\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Expand $$\\left(\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)\\cdot\\:2:{\\quad}2\\sec^{4}\\left(θ\\right)+4\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)$$", "input": "\\left(\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)\\cdot\\:2", "result": "=2\\sec^{4}\\left(θ\\right)+4\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)", "steps": [ { "type": "step", "result": "=2\\left(\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the distributive law: $$a\\left(b+c\\right)=ab+ac$$", "secondary": [ "$$a=2,\\:b=\\sec^{4}\\left(θ\\right),\\:c=2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)$$" ], "result": "=2\\sec^{4}\\left(θ\\right)+2\\cdot\\:2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=2\\sec^{4}\\left(θ\\right)+4\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7cTiUO1fOFax2wXHZewjhxEUAdh/O2PLAxzMjSpKce9YL4tWjzbYCZhBh/Hc2rUXacMKakGXZAry+N/hvJOodjXCQoYlYQ8U+Tfyx0kyzI8iF3F2TkMs7mKjIsUPAR/Vd82dH/qvnCC7/oZy4HZlX2aHh6acbPp8G70jj/skoFx7vbBmbuQNTF0TphKZ8Ruvacu+LCkBvFQsw7xBL2Ys2TUYb1899n2cPVSJTSzzdZzp+38s+uQz1W6FLDk3KkIYWhJuRFkG6Ai089lDjeB1lMbCI2sSeA74029n2yo277ZU=" } }, { "type": "interim", "title": "Rewrite using trig identities", "input": "2\\sec^{4}\\left(θ\\right)+4\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)", "result": "=-4\\sec^{2}\\left(θ\\right)+6\\sec^{4}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Use the Pythagorean identity: $$\\tan^{2}\\left(x\\right)+1=\\sec^{2}\\left(x\\right)$$", "secondary": [ "$$\\tan^{2}\\left(x\\right)=\\sec^{2}\\left(x\\right)-1$$" ], "result": "=2\\sec^{4}\\left(θ\\right)+4\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right)" }, { "type": "interim", "title": "Simplify $$2\\sec^{4}\\left(θ\\right)+4\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right):{\\quad}6\\sec^{4}\\left(θ\\right)-4\\sec^{2}\\left(θ\\right)$$", "input": "2\\sec^{4}\\left(θ\\right)+4\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right)", "result": "=6\\sec^{4}\\left(θ\\right)-4\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "interim", "title": "Expand $$4\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right):{\\quad}4\\sec^{4}\\left(θ\\right)-4\\sec^{2}\\left(θ\\right)$$", "input": "4\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right)", "result": "=2\\sec^{4}\\left(θ\\right)+4\\sec^{4}\\left(θ\\right)-4\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b-c\\right)=ab-ac$$", "secondary": [ "$$a=4\\sec^{2}\\left(θ\\right),\\:b=\\sec^{2}\\left(θ\\right),\\:c=1$$" ], "result": "=4\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)-4\\sec^{2}\\left(θ\\right)\\cdot\\:1", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "result": "=4\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)-4\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$4\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)-4\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right):{\\quad}4\\sec^{4}\\left(θ\\right)-4\\sec^{2}\\left(θ\\right)$$", "input": "4\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)-4\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right)", "result": "=4\\sec^{4}\\left(θ\\right)-4\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "interim", "title": "$$4\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)=4\\sec^{4}\\left(θ\\right)$$", "input": "4\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)=\\:\\sec^{2+2}\\left(θ\\right)$$" ], "result": "=4\\sec^{2+2}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+2=4$$", "result": "=4\\sec^{4}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vjDr4p6TvcgiSLV1uDo4U9JYPDPDesonRI520G8zYRPehkKrn0era9rz8TlL+x/vMHVZf8xiryCPkbUIYXhiIHRfFgXB+vlRb/AHCJklpF9e7YF3TFAJSjcLX2VvhbubqX/cSuhK5Ty2xqd/IdNr6zlZ8CVoKZl2BAoDdAtcFNxU2fg2Eo8khoQfnK/P3ock" } }, { "type": "interim", "title": "$$4\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right)=4\\sec^{2}\\left(θ\\right)$$", "input": "4\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:1=4$$", "result": "=4\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7xF9rlQkuAOrRzKTf6eQs7fgkIqVBCzsTrV5i/ECNCw8tOtZYwUjyXhDTsNnn6Elr/BCsllMMEyc8UGKoUwuYxu6vfsn/ogOG+fHMfNQh9JeZfoRPsbCjih+xtMTOMGl4tzgLeG7CLgj8aHpc44iME/LWpEthEIQrAjNHY4qDKt/ET1Ehq4mhUZ9DWVx2az0M" } }, { "type": "step", "result": "=4\\sec^{4}\\left(θ\\right)-4\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vjDr4p6TvcgiSLV1uDo4U/jbMbEoNNbIR/7Q84OynCfTLx8mOdHYVzxX643JqKFIvTZkYD4gUeSoFrhGGhxq98kwYX4VRPb1geKkn+NU7mcE3SiHEhDlxIXJDVY+PPdvgQUxJPyUNnGfVirkcwpVO84fwxlvni/T8DIWnD0mw9U6FFV/0Hm6Y1hI7VaPzzKhkSoZP7CyArJlOo0FRny79Q==" } }, { "type": "step", "primary": "Add similar elements: $$2\\sec^{4}\\left(θ\\right)+4\\sec^{4}\\left(θ\\right)=6\\sec^{4}\\left(θ\\right)$$", "result": "=6\\sec^{4}\\left(θ\\right)-4\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7f48emNsIl4MI4hQZuZje4penFTHzh6X+DX3uvPdqYy7umM/GfBKdAtyuCOAeIKZmAJYpRu9XpYrd8NSAW2DdD0c/CXknyRKFbCevkP3vkuVyZXAw9UlCE0mEZtdsFPrV7q9+yf+iA4b58cx81CH0l/C30sSftAIFS6Qkpy19Ikp0x8irWzKvvppKMUvqTrnrUObZozYPTEmgGDR5V3grHDoUVX/QebpjWEjtVo/PMqGRKhk/sLICsmU6jQVGfLv1" } } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7xqDNX42OyJjPevbF+7IBng7708BkbVE1gb8/0kMEgzvXUIQv0aG2ncbDgSWX/DWycNZx/2EDub2dSsLw3NGg9xLC2T8NFZvTA+pff5AdXPdX9lRN4mlYq/zI0dgxEXGdPvdnhi0s3lZ+DJ7rEiTeqenxKTmAB1ahPFE542Dd/TeutOLi0kB5xHysaBsHIgntIaTy8gmKy5PY3el+RbUmlkmFP65WPUHCIGTpzT/EAccetg60+ZUlqP5jB+ReldUA5AMRxXWt+t9va0qZyagxyQ==" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7xI7dknI212svHgZHHIcoRvkdSragnZ5K8+WMRM5qjJ6ETkUamdUz59t0aHpV3uN0rE0Ib0rKQ2oM97vl/gDbowCWKUbvV6WK3fDUgFtg3Q+MfZ1BFRhhGnQf9q28g9nK/SzNF20Xrftd883pI0RALQ5+VkRmueXOP55Piqx5MyWBBTEk/JQ2cZ9WKuRzClU7m4RppR4RRNJR6+bqv/lHExUnakC9Y1jhr4uI97E9yvalDk6K3L00/zgNSSliALTtnpy3tAot7+5PIP9fBKZw2Llxun299aFzbjCyRFblUKE=" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{d}{dθ}\\left(-4\\sec^{2}\\left(θ\\right)+6\\sec^{4}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(-4\\sec^{2}\\left(θ\\right)+6\\sec^{4}\\left(θ\\right)\\right)=-8\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+24\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(-4\\sec^{2}\\left(θ\\right)+6\\sec^{4}\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=-\\frac{d}{dθ}\\left(4\\sec^{2}\\left(θ\\right)\\right)+\\frac{d}{dθ}\\left(6\\sec^{4}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(4\\sec^{2}\\left(θ\\right)\\right)=8\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(4\\sec^{2}\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=4\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Apply the chain rule:$${\\quad}2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{2},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{2}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{2}\\right)=2u$$", "input": "\\frac{d}{du}\\left(u^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2u^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2u", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYr+VZUwBnLdzbS6DZQ+f4s+k3hxk9aCfAWodBRxXgUexMchyqTAJWrzJaDbnNcFsJUeCBKuYKgaNJ253gLI69U79qbCA2QqVmvm3jGRXZ2ppvbGT4j1utMEkCDH25m/vlQ==" } }, { "type": "step", "result": "=2u\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgCSXIThcWH58oYoMPt18lzFqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidZClirlWNFPnHR1rvdyoFLI7IC+oqnwb1T6nPLpiw1eysw2NXPd8jZnr3fcPpqc7ejeh7+jKEzLb7VNCEMF3Z/Maswa5oIagKQFOfrnKm5+Gbb/PxX3awut5KxRxyZf8A=" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=4\\cdot\\:2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$4\\cdot\\:2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}8\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "4\\cdot\\:2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=8\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:2=8$$", "result": "=8\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{1+1}\\left(θ\\right)$$" ], "result": "=8\\sec^{1+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=8\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7NqBYpvGKZC5lMRQos2qph5K/zREn/U8meKQ3TvRjTQz8cZMsH3IX6lxp+ANHGbuCq47vuWedXv2WUg94ER8IwTfgzf+5OuhkULh+nunjrqae0Rv083m3K3ie3DVArOu08LfSxJ+0AgVLpCSnLX0iSqXiJoCK22T5YMsRbcSz8DsyFZaieHAvmtaw5r6RKguDHxgfR9uWXPlGtc+ysPJIG+IASZeFjDtawNGt9P21GJY=" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(6\\sec^{4}\\left(θ\\right)\\right)=24\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(6\\sec^{4}\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=6\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Apply the chain rule:$${\\quad}4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{4},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{4}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{4}\\right)=4u^{3}$$", "input": "\\frac{d}{du}\\left(u^{4}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=4u^{4-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=4u^{3}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYl3oEe4EC6n4B8Xgz0cKnkqk3hxk9aCfAWodBRxXgUexuzo0Fl6VhneutEteidn+Kv8//6/nV5O4fb8Xgwi7mapNFxUvwBeni+JEIFAdbegzYEOcQgyW4n51cR7AeMAm7Q==" } }, { "type": "step", "result": "=4u^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgA2CO5cm49xNEqFV0DMaZJRqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidVduWEnoDYDg06IbaTsp9TFtqiuNhFTvmF8ZCDqWZNoubfnOah8gAvLMKOeNMctC5tVYZ/3r82LxPimKUFXjQFStvD8GKDNJJN8qiu0XRcVZAYAvbRKqkRVDaDBHkSBtDwYHkfmHHTze3JXk3LjiJp" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=6\\cdot\\:4\\left(\\sec\\left(θ\\right)\\right)^{3}\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$6\\cdot\\:4\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}24\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "6\\cdot\\:4\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=24\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$6\\cdot\\:4=24$$", "result": "=24\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{3+1}\\left(θ\\right)$$" ], "result": "=24\\sec^{3+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$3+1=4$$", "result": "=24\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l3FkS70/4BFsebxcaZbHnTNSoP5XQt+n/5y371e8QHJROlTYN926lBwmiJdD5HUSzRqDxPUzBN6vjj5oJL9kUIdu2MPseykkfgn/Trt10KGp9ej8WEAztP3UNEC9n7jjZEt3ZXAiqUE0HIXrrrezJHZKl2mV4AlztiApTAFigz07AW6Jz0sXkPNp6b6xpHMqM1Kg/ldC36f/nLfvV7xActytFlzCV8oq55bHLwXbb0c=" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=-8\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+24\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=-8\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+24\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "Evaluate $$-8\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+24\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\:$$at point $$θ=0:{\\quad}0$$", "result": "=0", "steps": [ { "type": "step", "primary": "Take the point $$θ=0$$ and plug it into $$-8\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+24\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=-8\\sec^{2}\\left(0\\right)\\tan\\left(0\\right)+24\\sec^{4}\\left(0\\right)\\tan\\left(0\\right)" }, { "type": "interim", "title": "$$8\\sec^{2}\\left(0\\right)\\tan\\left(0\\right)=0$$", "input": "8\\sec^{2}\\left(0\\right)\\tan\\left(0\\right)", "steps": [ { "type": "interim", "title": "$$\\sec^{2}\\left(0\\right)=1$$", "input": "\\sec^{2}\\left(0\\right)", "steps": [ { "type": "interim", "title": "Simplify $$\\sec\\left(0\\right):{\\quad}1$$", "input": "\\sec\\left(0\\right)", "result": "=1^{2}", "steps": [ { "type": "step", "primary": "Use the following trivial identity:$${\\quad}\\sec\\left(0\\right)=1$$", "secondary": [ "$$\\sec\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sec(x)&x&\\sec(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{2\\sqrt{3}}{3}&\\frac{7π}{6}&-\\frac{2\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&\\sqrt{2}&\\frac{5π}{4}&-\\sqrt{2}\\\\\\hline \\frac{π}{3}&2&\\frac{4π}{3}&-2\\\\\\hline \\frac{π}{2}&\\mathrm{Undefined}&\\frac{3π}{2}&\\mathrm{Undefined}\\\\\\hline \\frac{2π}{3}&-2&\\frac{5π}{3}&2\\\\\\hline \\frac{3π}{4}&-\\sqrt{2}&\\frac{7π}{4}&\\sqrt{2}\\\\\\hline \\frac{5π}{6}&-\\frac{2\\sqrt{3}}{3}&\\frac{11π}{6}&\\frac{2\\sqrt{3}}{3}\\\\\\hline \\end{array}$$" ], "result": "=1" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7y1xaZeQTSzp609voOcYk6VXTSum/z5kLpMzXS1UJIexiYFOaxxQqg3o8CNKkTm2qW+Sc57RV820557NJqaC71BfVjTo90KMWspnEdE4CKjE=" } }, { "type": "step", "result": "=8\\cdot\\:1\\cdot\\:\\tan\\left(0\\right)" }, { "type": "interim", "title": "Simplify $$\\tan\\left(0\\right):{\\quad}0$$", "input": "\\tan\\left(0\\right)", "result": "=8\\cdot\\:1\\cdot\\:0", "steps": [ { "type": "step", "primary": "Use the following trivial identity:$${\\quad}\\tan\\left(0\\right)=0$$", "secondary": [ "$$\\tan\\left(x\\right)$$ periodicity table with $$πn$$ cycle:<br/>$$\\begin{array}{|c|c|}\\hline x&\\tan(x)\\\\\\hline 0&0\\\\\\hline \\frac{π}{6}&\\frac{\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&1\\\\\\hline \\frac{π}{3}&\\sqrt{3}\\\\\\hline \\frac{π}{2}&\\pm\\infty\\\\\\hline \\frac{2π}{3}&-\\sqrt{3}\\\\\\hline \\frac{3π}{4}&-1\\\\\\hline \\frac{5π}{6}&-\\frac{\\sqrt{3}}{3}\\\\\\hline &\\\\\\hline \\end{array}$$" ], "result": "=0" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Apply rule $$0\\cdot\\:a=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7VXkP8RYre4Z4DcXgpt1XXqmPj3gXRowY53lqkfZS9MsJQJZuTAY5js+oqjdT8kslKXPrgUnq5rRq9Cvw1ceDcuYFtoJybdMeJwh0lzmf693zoXLK02yt5K+3YXszYC4c" } }, { "type": "interim", "title": "$$24\\sec^{4}\\left(0\\right)\\tan\\left(0\\right)=0$$", "input": "24\\sec^{4}\\left(0\\right)\\tan\\left(0\\right)", "steps": [ { "type": "interim", "title": "$$\\sec^{4}\\left(0\\right)=1$$", "input": "\\sec^{4}\\left(0\\right)", "steps": [ { "type": "interim", "title": "Simplify $$\\sec\\left(0\\right):{\\quad}1$$", "input": "\\sec\\left(0\\right)", "result": "=1^{4}", "steps": [ { "type": "step", "primary": "Use the following trivial identity:$${\\quad}\\sec\\left(0\\right)=1$$", "secondary": [ "$$\\sec\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sec(x)&x&\\sec(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{2\\sqrt{3}}{3}&\\frac{7π}{6}&-\\frac{2\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&\\sqrt{2}&\\frac{5π}{4}&-\\sqrt{2}\\\\\\hline \\frac{π}{3}&2&\\frac{4π}{3}&-2\\\\\\hline \\frac{π}{2}&\\mathrm{Undefined}&\\frac{3π}{2}&\\mathrm{Undefined}\\\\\\hline \\frac{2π}{3}&-2&\\frac{5π}{3}&2\\\\\\hline \\frac{3π}{4}&-\\sqrt{2}&\\frac{7π}{4}&\\sqrt{2}\\\\\\hline \\frac{5π}{6}&-\\frac{2\\sqrt{3}}{3}&\\frac{11π}{6}&\\frac{2\\sqrt{3}}{3}\\\\\\hline \\end{array}$$" ], "result": "=1" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7TnbPBMsEd78kmmwfYDHqNVXTSum/z5kLpMzXS1UJIexiYFOaxxQqg3o8CNKkTm2qW+Sc57RV820557NJqaC71Ka6Ma3rg33eDIhfX/iuVnM=" } }, { "type": "step", "result": "=24\\cdot\\:1\\cdot\\:\\tan\\left(0\\right)" }, { "type": "interim", "title": "Simplify $$\\tan\\left(0\\right):{\\quad}0$$", "input": "\\tan\\left(0\\right)", "result": "=24\\cdot\\:1\\cdot\\:0", "steps": [ { "type": "step", "primary": "Use the following trivial identity:$${\\quad}\\tan\\left(0\\right)=0$$", "secondary": [ "$$\\tan\\left(x\\right)$$ periodicity table with $$πn$$ cycle:<br/>$$\\begin{array}{|c|c|}\\hline x&\\tan(x)\\\\\\hline 0&0\\\\\\hline \\frac{π}{6}&\\frac{\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&1\\\\\\hline \\frac{π}{3}&\\sqrt{3}\\\\\\hline \\frac{π}{2}&\\pm\\infty\\\\\\hline \\frac{2π}{3}&-\\sqrt{3}\\\\\\hline \\frac{3π}{4}&-1\\\\\\hline \\frac{5π}{6}&-\\frac{\\sqrt{3}}{3}\\\\\\hline &\\\\\\hline \\end{array}$$" ], "result": "=0" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Apply rule $$0\\cdot\\:a=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s78VDcKUNKscENXvvWg21p8xjTuW+Q2J5Q5LUQGIymb9t1g99dC9fj9sg0EHzBIRDRd79UrkSVT0SCLs80Lgihl2Q3g/ebOMbnBgVLW2ni2Bnbzo7SL2lLjXVd+4Xv52R0JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=-0+0" }, { "type": "step", "primary": "Simplify" }, { "type": "step", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Laplace Eval Function At Point Specific 3Eq" } }, { "type": "step", "result": "0" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\frac{d^{5}}{dθ^{5}}\\left(\\tan\\left(θ\\right)\\right)\\left(0\\right){\\quad:\\quad}16$$", "input": "\\frac{d^{5}}{dθ^{5}}\\left(\\tan\\left(θ\\right)\\right)\\left(0\\right)", "steps": [ { "type": "interim", "title": "$$\\frac{d^{5}}{dθ^{5}}\\left(\\tan\\left(θ\\right)\\right)=-8\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)+24\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)$$", "input": "\\frac{d^{5}}{dθ^{5}}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "result": "=\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYs0Vsx3c0nt53Utu5DgHTe3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK00HV/bLwwglXz/2YNx2i4JHggSrmCoGjSdud4CyOvVOgdQN0k+N3NHNAkEIoLrhB631k80nn+ODwI4ID0kCF5+z7zRz81SqRD02HEUE4Wqh" } }, { "type": "step", "result": "=\\frac{d^{4}}{dθ^{4}}\\left(\\sec^{2}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{2},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{2}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{2}\\right)=2u$$", "input": "\\frac{d}{du}\\left(u^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2u^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2u", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYr+VZUwBnLdzbS6DZQ+f4s+k3hxk9aCfAWodBRxXgUexMchyqTAJWrzJaDbnNcFsJUeCBKuYKgaNJ253gLI69U79qbCA2QqVmvm3jGRXZ2ppvbGT4j1utMEkCDH25m/vlQ==" } }, { "type": "step", "result": "=2u\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgCSXIThcWH58oYoMPt18lzFqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidZClirlWNFPnHR1rvdyoFLI7IC+oqnwb1T6nPLpiw1eysw2NXPd8jZnr3fcPpqc7ejeh7+jKEzLb7VNCEMF3Z/Maswa5oIagKQFOfrnKm5+Gbb/PxX3awut5KxRxyZf8A=" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{1+1}\\left(θ\\right)$$" ], "result": "=2\\sec^{1+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7aY4rYekMZhUfP6sZ6mfqd6Lsz0lt1XLEr88Kc/cjJo5V00rpv8+ZC6TM10tVCSHspqrCcTwb/I0f8OwMjuU6eBAUa+AHsHd5OGpMiuFITjOuqvW3hAHH635V/vEEYIatHimBRYRqHSWeJkuUPhfTC5LQ0B1JNgsb5SXzbS+SCnAfGB9H25Zc+Ua1z7Kw8kgb4gBJl4WMO1rA0a30/bUYlg==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{d^{3}}{dθ^{3}}\\left(2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\right)=-4\\sec^{2}\\left(θ\\right)+6\\sec^{4}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=2\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the Product Rule: $$\\left(f{\\cdot}g\\right)'=f'{\\cdot}g+f{\\cdot}g'$$", "secondary": [ "$$f=\\sec^{2}\\left(θ\\right),\\:g=\\tan\\left(θ\\right)$$" ], "result": "=2\\left(\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)\\tan\\left(θ\\right)+\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)\\sec^{2}\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Product%20Rule", "practiceTopic": "Product Rule" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{2},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{2}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{2}\\right)=2u$$", "input": "\\frac{d}{du}\\left(u^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2u^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2u", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYr+VZUwBnLdzbS6DZQ+f4s+k3hxk9aCfAWodBRxXgUexMchyqTAJWrzJaDbnNcFsJUeCBKuYKgaNJ253gLI69U79qbCA2QqVmvm3jGRXZ2ppvbGT4j1utMEkCDH25m/vlQ==" } }, { "type": "step", "result": "=2u\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgCSXIThcWH58oYoMPt18lzFqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidZClirlWNFPnHR1rvdyoFLI7IC+oqnwb1T6nPLpiw1eysw2NXPd8jZnr3fcPpqc7ejeh7+jKEzLb7VNCEMF3Z/Maswa5oIagKQFOfrnKm5+Gbb/PxX3awut5KxRxyZf8A=" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{1+1}\\left(θ\\right)$$" ], "result": "=2\\sec^{1+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7aY4rYekMZhUfP6sZ6mfqd6Lsz0lt1XLEr88Kc/cjJo5V00rpv8+ZC6TM10tVCSHspqrCcTwb/I0f8OwMjuU6eBAUa+AHsHd5OGpMiuFITjOuqvW3hAHH635V/vEEYIatHimBRYRqHSWeJkuUPhfTC5LQ0B1JNgsb5SXzbS+SCnAfGB9H25Zc+Ua1z7Kw8kgb4gBJl4WMO1rA0a30/bUYlg==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "result": "=\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYs0Vsx3c0nt53Utu5DgHTe3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK00HV/bLwwglXz/2YNx2i4JHggSrmCoGjSdud4CyOvVOgdQN0k+N3NHNAkEIoLrhB631k80nn+ODwI4ID0kCF5+z7zRz81SqRD02HEUE4Wqh" } }, { "type": "step", "result": "=2\\left(2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Simplify $$2\\left(2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\right):{\\quad}-4\\sec^{2}\\left(θ\\right)+6\\sec^{4}\\left(θ\\right)$$", "input": "2\\left(2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\right)", "result": "=-4\\sec^{2}\\left(θ\\right)+6\\sec^{4}\\left(θ\\right)", "steps": [ { "type": "interim", "title": "$$2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)=2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)$$", "input": "2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\tan\\left(θ\\right)\\tan\\left(θ\\right)=\\:\\tan^{1+1}\\left(θ\\right)$$" ], "result": "=2\\sec^{2}\\left(θ\\right)\\tan^{1+1}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Md1SJKHYLxhOYRRbsLTu+axUZgt6FTmFSHnRsotkqemLeJKmeakO1+Q3aSIldG41s7BwSVdKEYSqn+4fKAQoGk0xmFIl1KTO5aAkhaHvKzBALhShbMGZIZsZWTO721HsWI4dG63F/aSz/PFDvu5FnO3sngYNzcydIedGQjjtM8sBMUhkLMqDD6KHvgZAgKMdRQzIIgXxnWeDhDTk2Lfum8RPUSGriaFRn0NZXHZrPQw=" } }, { "type": "interim", "title": "$$\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)=\\sec^{4}\\left(θ\\right)$$", "input": "\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)=\\:\\sec^{2+2}\\left(θ\\right)$$" ], "result": "=\\sec^{2+2}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+2=4$$", "result": "=\\sec^{4}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7AtrsReczPdUqIdn5/Ev5mFtNfVJqszzpMwg6spHO8F8gJ/ZZA32ZInFBpDtxBfiKn3Jyj2VJgBojKQHjINpkkETYvnjqwhFu6aWmDOYpAVMO/U6qyRZonn1WPQ/21pz3cKXVLLQXMot2gtnZH7qY5zkbLioo7YVO0SRnDAO/7Uokt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "=2\\left(\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Expand $$\\left(\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)\\cdot\\:2:{\\quad}2\\sec^{4}\\left(θ\\right)+4\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)$$", "input": "\\left(\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)\\cdot\\:2", "result": "=2\\sec^{4}\\left(θ\\right)+4\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)", "steps": [ { "type": "step", "result": "=2\\left(\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the distributive law: $$a\\left(b+c\\right)=ab+ac$$", "secondary": [ "$$a=2,\\:b=\\sec^{4}\\left(θ\\right),\\:c=2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)$$" ], "result": "=2\\sec^{4}\\left(θ\\right)+2\\cdot\\:2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=2\\sec^{4}\\left(θ\\right)+4\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7cTiUO1fOFax2wXHZewjhxEUAdh/O2PLAxzMjSpKce9YL4tWjzbYCZhBh/Hc2rUXacMKakGXZAry+N/hvJOodjXCQoYlYQ8U+Tfyx0kyzI8iF3F2TkMs7mKjIsUPAR/Vd82dH/qvnCC7/oZy4HZlX2aHh6acbPp8G70jj/skoFx7vbBmbuQNTF0TphKZ8Ruvacu+LCkBvFQsw7xBL2Ys2TUYb1899n2cPVSJTSzzdZzp+38s+uQz1W6FLDk3KkIYWhJuRFkG6Ai089lDjeB1lMbCI2sSeA74029n2yo277ZU=" } }, { "type": "interim", "title": "Rewrite using trig identities", "input": "2\\sec^{4}\\left(θ\\right)+4\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)", "result": "=-4\\sec^{2}\\left(θ\\right)+6\\sec^{4}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Use the Pythagorean identity: $$\\tan^{2}\\left(x\\right)+1=\\sec^{2}\\left(x\\right)$$", "secondary": [ "$$\\tan^{2}\\left(x\\right)=\\sec^{2}\\left(x\\right)-1$$" ], "result": "=2\\sec^{4}\\left(θ\\right)+4\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right)" }, { "type": "interim", "title": "Simplify $$2\\sec^{4}\\left(θ\\right)+4\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right):{\\quad}6\\sec^{4}\\left(θ\\right)-4\\sec^{2}\\left(θ\\right)$$", "input": "2\\sec^{4}\\left(θ\\right)+4\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right)", "result": "=6\\sec^{4}\\left(θ\\right)-4\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "interim", "title": "Expand $$4\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right):{\\quad}4\\sec^{4}\\left(θ\\right)-4\\sec^{2}\\left(θ\\right)$$", "input": "4\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right)", "result": "=2\\sec^{4}\\left(θ\\right)+4\\sec^{4}\\left(θ\\right)-4\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b-c\\right)=ab-ac$$", "secondary": [ "$$a=4\\sec^{2}\\left(θ\\right),\\:b=\\sec^{2}\\left(θ\\right),\\:c=1$$" ], "result": "=4\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)-4\\sec^{2}\\left(θ\\right)\\cdot\\:1", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "result": "=4\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)-4\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$4\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)-4\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right):{\\quad}4\\sec^{4}\\left(θ\\right)-4\\sec^{2}\\left(θ\\right)$$", "input": "4\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)-4\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right)", "result": "=4\\sec^{4}\\left(θ\\right)-4\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "interim", "title": "$$4\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)=4\\sec^{4}\\left(θ\\right)$$", "input": "4\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)=\\:\\sec^{2+2}\\left(θ\\right)$$" ], "result": "=4\\sec^{2+2}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+2=4$$", "result": "=4\\sec^{4}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vjDr4p6TvcgiSLV1uDo4U9JYPDPDesonRI520G8zYRPehkKrn0era9rz8TlL+x/vMHVZf8xiryCPkbUIYXhiIHRfFgXB+vlRb/AHCJklpF9e7YF3TFAJSjcLX2VvhbubqX/cSuhK5Ty2xqd/IdNr6zlZ8CVoKZl2BAoDdAtcFNxU2fg2Eo8khoQfnK/P3ock" } }, { "type": "interim", "title": "$$4\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right)=4\\sec^{2}\\left(θ\\right)$$", "input": "4\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:1=4$$", "result": "=4\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7xF9rlQkuAOrRzKTf6eQs7fgkIqVBCzsTrV5i/ECNCw8tOtZYwUjyXhDTsNnn6Elr/BCsllMMEyc8UGKoUwuYxu6vfsn/ogOG+fHMfNQh9JeZfoRPsbCjih+xtMTOMGl4tzgLeG7CLgj8aHpc44iME/LWpEthEIQrAjNHY4qDKt/ET1Ehq4mhUZ9DWVx2az0M" } }, { "type": "step", "result": "=4\\sec^{4}\\left(θ\\right)-4\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vjDr4p6TvcgiSLV1uDo4U/jbMbEoNNbIR/7Q84OynCfTLx8mOdHYVzxX643JqKFIvTZkYD4gUeSoFrhGGhxq98kwYX4VRPb1geKkn+NU7mcE3SiHEhDlxIXJDVY+PPdvgQUxJPyUNnGfVirkcwpVO84fwxlvni/T8DIWnD0mw9U6FFV/0Hm6Y1hI7VaPzzKhkSoZP7CyArJlOo0FRny79Q==" } }, { "type": "step", "primary": "Add similar elements: $$2\\sec^{4}\\left(θ\\right)+4\\sec^{4}\\left(θ\\right)=6\\sec^{4}\\left(θ\\right)$$", "result": "=6\\sec^{4}\\left(θ\\right)-4\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7f48emNsIl4MI4hQZuZje4penFTHzh6X+DX3uvPdqYy7umM/GfBKdAtyuCOAeIKZmAJYpRu9XpYrd8NSAW2DdD0c/CXknyRKFbCevkP3vkuVyZXAw9UlCE0mEZtdsFPrV7q9+yf+iA4b58cx81CH0l/C30sSftAIFS6Qkpy19Ikp0x8irWzKvvppKMUvqTrnrUObZozYPTEmgGDR5V3grHDoUVX/QebpjWEjtVo/PMqGRKhk/sLICsmU6jQVGfLv1" } } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7xqDNX42OyJjPevbF+7IBng7708BkbVE1gb8/0kMEgzvXUIQv0aG2ncbDgSWX/DWycNZx/2EDub2dSsLw3NGg9xLC2T8NFZvTA+pff5AdXPdX9lRN4mlYq/zI0dgxEXGdPvdnhi0s3lZ+DJ7rEiTeqenxKTmAB1ahPFE542Dd/TeutOLi0kB5xHysaBsHIgntIaTy8gmKy5PY3el+RbUmlkmFP65WPUHCIGTpzT/EAccetg60+ZUlqP5jB+ReldUA5AMRxXWt+t9va0qZyagxyQ==" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7xI7dknI212svHgZHHIcoRvkdSragnZ5K8+WMRM5qjJ6ETkUamdUz59t0aHpV3uN0rE0Ib0rKQ2oM97vl/gDbowCWKUbvV6WK3fDUgFtg3Q+MfZ1BFRhhGnQf9q28g9nK/SzNF20Xrftd883pI0RALQ5+VkRmueXOP55Piqx5MyWBBTEk/JQ2cZ9WKuRzClU7m4RppR4RRNJR6+bqv/lHExUnakC9Y1jhr4uI97E9yvalDk6K3L00/zgNSSliALTtnpy3tAot7+5PIP9fBKZw2Llxun299aFzbjCyRFblUKE=" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{d^{2}}{dθ^{2}}\\left(-4\\sec^{2}\\left(θ\\right)+6\\sec^{4}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(-4\\sec^{2}\\left(θ\\right)+6\\sec^{4}\\left(θ\\right)\\right)=-8\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+24\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(-4\\sec^{2}\\left(θ\\right)+6\\sec^{4}\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=-\\frac{d}{dθ}\\left(4\\sec^{2}\\left(θ\\right)\\right)+\\frac{d}{dθ}\\left(6\\sec^{4}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(4\\sec^{2}\\left(θ\\right)\\right)=8\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(4\\sec^{2}\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=4\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Apply the chain rule:$${\\quad}2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{2},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{2}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{2}\\right)=2u$$", "input": "\\frac{d}{du}\\left(u^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2u^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2u", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYr+VZUwBnLdzbS6DZQ+f4s+k3hxk9aCfAWodBRxXgUexMchyqTAJWrzJaDbnNcFsJUeCBKuYKgaNJ253gLI69U79qbCA2QqVmvm3jGRXZ2ppvbGT4j1utMEkCDH25m/vlQ==" } }, { "type": "step", "result": "=2u\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgCSXIThcWH58oYoMPt18lzFqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidZClirlWNFPnHR1rvdyoFLI7IC+oqnwb1T6nPLpiw1eysw2NXPd8jZnr3fcPpqc7ejeh7+jKEzLb7VNCEMF3Z/Maswa5oIagKQFOfrnKm5+Gbb/PxX3awut5KxRxyZf8A=" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=4\\cdot\\:2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$4\\cdot\\:2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}8\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "4\\cdot\\:2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=8\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:2=8$$", "result": "=8\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{1+1}\\left(θ\\right)$$" ], "result": "=8\\sec^{1+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=8\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7NqBYpvGKZC5lMRQos2qph5K/zREn/U8meKQ3TvRjTQz8cZMsH3IX6lxp+ANHGbuCq47vuWedXv2WUg94ER8IwTfgzf+5OuhkULh+nunjrqae0Rv083m3K3ie3DVArOu08LfSxJ+0AgVLpCSnLX0iSqXiJoCK22T5YMsRbcSz8DsyFZaieHAvmtaw5r6RKguDHxgfR9uWXPlGtc+ysPJIG+IASZeFjDtawNGt9P21GJY=" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(6\\sec^{4}\\left(θ\\right)\\right)=24\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(6\\sec^{4}\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=6\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Apply the chain rule:$${\\quad}4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{4},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{4}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{4}\\right)=4u^{3}$$", "input": "\\frac{d}{du}\\left(u^{4}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=4u^{4-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=4u^{3}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYl3oEe4EC6n4B8Xgz0cKnkqk3hxk9aCfAWodBRxXgUexuzo0Fl6VhneutEteidn+Kv8//6/nV5O4fb8Xgwi7mapNFxUvwBeni+JEIFAdbegzYEOcQgyW4n51cR7AeMAm7Q==" } }, { "type": "step", "result": "=4u^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgA2CO5cm49xNEqFV0DMaZJRqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidVduWEnoDYDg06IbaTsp9TFtqiuNhFTvmF8ZCDqWZNoubfnOah8gAvLMKOeNMctC5tVYZ/3r82LxPimKUFXjQFStvD8GKDNJJN8qiu0XRcVZAYAvbRKqkRVDaDBHkSBtDwYHkfmHHTze3JXk3LjiJp" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=6\\cdot\\:4\\left(\\sec\\left(θ\\right)\\right)^{3}\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$6\\cdot\\:4\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}24\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "6\\cdot\\:4\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=24\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$6\\cdot\\:4=24$$", "result": "=24\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{3+1}\\left(θ\\right)$$" ], "result": "=24\\sec^{3+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$3+1=4$$", "result": "=24\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l3FkS70/4BFsebxcaZbHnTNSoP5XQt+n/5y371e8QHJROlTYN926lBwmiJdD5HUSzRqDxPUzBN6vjj5oJL9kUIdu2MPseykkfgn/Trt10KGp9ej8WEAztP3UNEC9n7jjZEt3ZXAiqUE0HIXrrrezJHZKl2mV4AlztiApTAFigz07AW6Jz0sXkPNp6b6xpHMqM1Kg/ldC36f/nLfvV7xActytFlzCV8oq55bHLwXbb0c=" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=-8\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+24\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{d}{dθ}\\left(-8\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+24\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(-8\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+24\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)=-8\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)+24\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(-8\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+24\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=-\\frac{d}{dθ}\\left(8\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+\\frac{d}{dθ}\\left(24\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(8\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\right)=8\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(8\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=8\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the Product Rule: $$\\left(f{\\cdot}g\\right)'=f'{\\cdot}g+f{\\cdot}g'$$", "secondary": [ "$$f=\\sec^{2}\\left(θ\\right),\\:g=\\tan\\left(θ\\right)$$" ], "result": "=8\\left(\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)\\tan\\left(θ\\right)+\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)\\sec^{2}\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Product%20Rule", "practiceTopic": "Product Rule" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{2},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{2}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{2}\\right)=2u$$", "input": "\\frac{d}{du}\\left(u^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2u^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2u", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYr+VZUwBnLdzbS6DZQ+f4s+k3hxk9aCfAWodBRxXgUexMchyqTAJWrzJaDbnNcFsJUeCBKuYKgaNJ253gLI69U79qbCA2QqVmvm3jGRXZ2ppvbGT4j1utMEkCDH25m/vlQ==" } }, { "type": "step", "result": "=2u\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgCSXIThcWH58oYoMPt18lzFqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidZClirlWNFPnHR1rvdyoFLI7IC+oqnwb1T6nPLpiw1eysw2NXPd8jZnr3fcPpqc7ejeh7+jKEzLb7VNCEMF3Z/Maswa5oIagKQFOfrnKm5+Gbb/PxX3awut5KxRxyZf8A=" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{1+1}\\left(θ\\right)$$" ], "result": "=2\\sec^{1+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7aY4rYekMZhUfP6sZ6mfqd6Lsz0lt1XLEr88Kc/cjJo5V00rpv8+ZC6TM10tVCSHspqrCcTwb/I0f8OwMjuU6eBAUa+AHsHd5OGpMiuFITjOuqvW3hAHH635V/vEEYIatHimBRYRqHSWeJkuUPhfTC5LQ0B1JNgsb5SXzbS+SCnAfGB9H25Zc+Ua1z7Kw8kgb4gBJl4WMO1rA0a30/bUYlg==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "result": "=\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYs0Vsx3c0nt53Utu5DgHTe3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK00HV/bLwwglXz/2YNx2i4JHggSrmCoGjSdud4CyOvVOgdQN0k+N3NHNAkEIoLrhB631k80nn+ODwI4ID0kCF5+z7zRz81SqRD02HEUE4Wqh" } }, { "type": "step", "result": "=8\\left(2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Simplify $$8\\left(2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\right):{\\quad}8\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)$$", "input": "8\\left(2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\right)", "result": "=8\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "$$2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)=2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)$$", "input": "2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\tan\\left(θ\\right)\\tan\\left(θ\\right)=\\:\\tan^{1+1}\\left(θ\\right)$$" ], "result": "=2\\sec^{2}\\left(θ\\right)\\tan^{1+1}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Md1SJKHYLxhOYRRbsLTu+axUZgt6FTmFSHnRsotkqemLeJKmeakO1+Q3aSIldG41s7BwSVdKEYSqn+4fKAQoGk0xmFIl1KTO5aAkhaHvKzBALhShbMGZIZsZWTO721HsWI4dG63F/aSz/PFDvu5FnO3sngYNzcydIedGQjjtM8sBMUhkLMqDD6KHvgZAgKMdRQzIIgXxnWeDhDTk2Lfum8RPUSGriaFRn0NZXHZrPQw=" } }, { "type": "interim", "title": "$$\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)=\\sec^{4}\\left(θ\\right)$$", "input": "\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)=\\:\\sec^{2+2}\\left(θ\\right)$$" ], "result": "=\\sec^{2+2}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+2=4$$", "result": "=\\sec^{4}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7AtrsReczPdUqIdn5/Ev5mFtNfVJqszzpMwg6spHO8F8gJ/ZZA32ZInFBpDtxBfiKn3Jyj2VJgBojKQHjINpkkETYvnjqwhFu6aWmDOYpAVMO/U6qyRZonn1WPQ/21pz3cKXVLLQXMot2gtnZH7qY5zkbLioo7YVO0SRnDAO/7Uokt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "=8\\left(\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Simplify $$2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{4}\\left(θ\\right):{\\quad}3\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)$$", "input": "2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{4}\\left(θ\\right)", "result": "=8\\left(3\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Rewrite using trig identities", "input": "\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)", "result": "=-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Use the Pythagorean identity: $$\\tan^{2}\\left(x\\right)+1=\\sec^{2}\\left(x\\right)$$", "secondary": [ "$$\\tan^{2}\\left(x\\right)=\\sec^{2}\\left(x\\right)-1$$" ], "result": "=\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right)" }, { "type": "interim", "title": "Simplify $$\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right):{\\quad}3\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)$$", "input": "\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right)", "result": "=3\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "interim", "title": "Expand $$2\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right):{\\quad}2\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)$$", "input": "2\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right)", "result": "=\\sec^{4}\\left(θ\\right)+2\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b-c\\right)=ab-ac$$", "secondary": [ "$$a=2\\sec^{2}\\left(θ\\right),\\:b=\\sec^{2}\\left(θ\\right),\\:c=1$$" ], "result": "=2\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)\\cdot\\:1", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "result": "=2\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)-2\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$2\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)-2\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right):{\\quad}2\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)$$", "input": "2\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)-2\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right)", "result": "=2\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "interim", "title": "$$2\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)=2\\sec^{4}\\left(θ\\right)$$", "input": "2\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)=\\:\\sec^{2+2}\\left(θ\\right)$$" ], "result": "=2\\sec^{2+2}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+2=4$$", "result": "=2\\sec^{4}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Md1SJKHYLxhOYRRbsLTu+dJYPDPDesonRI520G8zYRPehkKrn0era9rz8TlL+x/vyvn14PieOyuPP98CkiPIaXRfFgXB+vlRb/AHCJklpF+qQC22PXYzQL8KzAhJCzszqX/cSuhK5Ty2xqd/IdNr67A2LXxYB8f3C3KpWaLgll1U2fg2Eo8khoQfnK/P3ock" } }, { "type": "interim", "title": "$$2\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right)=2\\sec^{2}\\left(θ\\right)$$", "input": "2\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=2\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ht1jgkupea8+jUgT+M/VFPgkIqVBCzsTrV5i/ECNCw8tOtZYwUjyXhDTsNnn6Elr8U8lPA4O6WXezAN1J2P7Ue6vfsn/ogOG+fHMfNQh9Je7opJQAzFYPpqn/2bLavXutzgLeG7CLgj8aHpc44iMEx91+u64Nzpp2/441mWGcQ/ET1Ehq4mhUZ9DWVx2az0M" } }, { "type": "step", "result": "=2\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Md1SJKHYLxhOYRRbsLTu+fjbMbEoNNbIR/7Q84OynCfTLx8mOdHYVzxX643JqKFI1Jetp9AvoQcY2QSDwFJhz1kUvGSawlk7umQ8kurx5doE3SiHEhDlxIXJDVY+PPdvgQUxJPyUNnGfVirkcwpVO6RMkE3rByM/Y1Dxxfs3xFw6FFV/0Hm6Y1hI7VaPzzKhkSoZP7CyArJlOo0FRny79Q==" } }, { "type": "step", "primary": "Add similar elements: $$\\sec^{4}\\left(θ\\right)+2\\sec^{4}\\left(θ\\right)=3\\sec^{4}\\left(θ\\right)$$", "result": "=3\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ARkhlGZdazrdxNLzhlqePtRiXXTzenxbt00ahL9sDS5L0MjjUe01mdeSwRMoYPSBLTrWWMFI8l4Q07DZ5+hJazaxvejhvTJvqoS105+T9+xc3qH/hVuCay9igw/1IwZBNySLUGdPDJeW3uY1XSs479bA+zX4bD3u3gx65o2NJhP8KqIsxz0lwScBMkZS31XhRCRwG0yFaMyhS3uC5/uUeMvqafR00cq/gMXQBnztqPn0yTzBsNnlyCSr/zrcxdQG" } } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7FkSOkw7Nc+ISjlBZ7XiESO/dTLaj0YrNSw6+/ogyGX8k+jBZ4XxujMMeS3Qvsi6m7aEe0hvT9VtxXei653t2zQi9Vnn4imguIoWn3N1eqjdGIkb3QW2w7y4N83RSktbc/kUupyp7xXxB8YmKEQ5HoPs4SAUa1zTfp2gkDWnxoYVtNtNgAtA4xHxju0w39zFkZEt3ZXAiqUE0HIXrrrezJJi0cdkgnw2HPjRVpcHMxZbIciYVzjaAolEaNkVONtBBRppTNgMFons9bMkTzIHooA==" } } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7JzuoJyZ0ket+0by442o6nfkdSragnZ5K8+WMRM5qjJ6ETkUamdUz59t0aHpV3uN0rE0Ib0rKQ2oM97vl/gDbowCWKUbvV6WK3fDUgFtg3Q+nr1JqgQ+t+tehVxpQusgo+5k7zGGz9wmA2lyQsVWy+j6F2h9EjPuS9/S594y1Nd1FKk3fejFkyiOiq9iG9IkAIuSmJUDtLxRPyJ57Nkq7qjUyCLMvQZ782OaKpAVh7FOsVGYLehU5hUh50bKLZKnpo1d4a+pBCGznOTLxP2sni3enTnyZLeuZs6pBwuQMplUkt3WiGR7ZaCaXvz77bMjS" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(24\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)=24\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(24\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=24\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the Product Rule: $$\\left(f{\\cdot}g\\right)'=f'{\\cdot}g+f{\\cdot}g'$$", "secondary": [ "$$f=\\sec^{4}\\left(θ\\right),\\:g=\\tan\\left(θ\\right)$$" ], "result": "=24\\left(\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)\\tan\\left(θ\\right)+\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)\\sec^{4}\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Product%20Rule", "practiceTopic": "Product Rule" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)=4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{4},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{4}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{4}\\right)=4u^{3}$$", "input": "\\frac{d}{du}\\left(u^{4}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=4u^{4-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=4u^{3}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYl3oEe4EC6n4B8Xgz0cKnkqk3hxk9aCfAWodBRxXgUexuzo0Fl6VhneutEteidn+Kv8//6/nV5O4fb8Xgwi7mapNFxUvwBeni+JEIFAdbegzYEOcQgyW4n51cR7AeMAm7Q==" } }, { "type": "step", "result": "=4u^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgA2CO5cm49xNEqFV0DMaZJRqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidVduWEnoDYDg06IbaTsp9TFtqiuNhFTvmF8ZCDqWZNoubfnOah8gAvLMKOeNMctC5tVYZ/3r82LxPimKUFXjQFStvD8GKDNJJN8qiu0XRcVZAYAvbRKqkRVDaDBHkSBtDwYHkfmHHTze3JXk3LjiJp" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$4\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "4\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{3+1}\\left(θ\\right)$$" ], "result": "=4\\sec^{3+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$3+1=4$$", "result": "=4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7p9w76+/nvHitDrMIupHrrrPmAPxuc/RFHWXkB8A+mnqLeJKmeakO1+Q3aSIldG41PCyA7ixjCdfDYe+yac7KOvOIrkYdKW3Q58EnXaMU+7oslHPSvPf5UfEs5T+IFZ4XHjb2+5NLFZrsH9fcPWg/TXL71sHPAsAQtYZvjpaGSRwboK/Igc1IRC3EGruCoRzV1Zrc/SYRIR8JuxYpD+PJ+Q==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "result": "=\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYs0Vsx3c0nt53Utu5DgHTe3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK00HV/bLwwglXz/2YNx2i4JHggSrmCoGjSdud4CyOvVOgdQN0k+N3NHNAkEIoLrhB631k80nn+ODwI4ID0kCF5+z7zRz81SqRD02HEUE4Wqh" } }, { "type": "step", "result": "=24\\left(4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Simplify $$24\\left(4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right):{\\quad}24\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)$$", "input": "24\\left(4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right)", "result": "=24\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "$$4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)=4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)$$", "input": "4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\tan\\left(θ\\right)\\tan\\left(θ\\right)=\\:\\tan^{1+1}\\left(θ\\right)$$" ], "result": "=4\\sec^{4}\\left(θ\\right)\\tan^{1+1}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7GHerUg4zB8o0EeBO6BKV16xUZgt6FTmFSHnRsotkqemLeJKmeakO1+Q3aSIldG41PCyA7ixjCdfDYe+yac7KOvOIrkYdKW3Q58EnXaMU+7pALhShbMGZIZsZWTO721HsXaBT4KNKuhZZSFsFWnKRStdNJOkRdCtrWB4iMQVmMwBoAWE3NiWlLPrDNnXLOZLK3tKI9zGyi4xCTaccmboyHcRPUSGriaFRn0NZXHZrPQw=" } }, { "type": "interim", "title": "$$\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)=\\sec^{6}\\left(θ\\right)$$", "input": "\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)=\\:\\sec^{2+4}\\left(θ\\right)$$" ], "result": "=\\sec^{2+4}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+4=6$$", "result": "=\\sec^{6}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7AtrsReczPdUqIdn5/Ev5mLBxohAhN47qWD2nlwOHqfkgJ/ZZA32ZInFBpDtxBfiKn3Jyj2VJgBojKQHjINpkkG6JHhxn2McSW7OxLgQBi64O/U6qyRZonn1WPQ/21pz3xMe3sR8ZMoA2Ynlp2W853t3OcQiSW6rZ4zI81UhADI4kt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "=24\\left(\\sec^{6}\\left(θ\\right)+4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7jhDHfABKmSqcJJ1fAFAsGBiMJmjxigW8LortZKvrqEbWPF6oWuXggupsFvXrooIqxMe3sR8ZMoA2Ynlp2W853tMvHyY50dhXPFfrjcmooUg3CXpyb+Iu3cMrCn545PU8uO1NenZJP60hEwBH05kmlvOIrkYdKW3Q58EnXaMU+7r4ZAjrJM4mwvgfYovsxoopgQUxJPyUNnGfVirkcwpVOxKnUVO0TgjLT50Mmk0DzLYh0PvFonSnuZi7Z6ItIhmYhYbzXwn7iok10o2w+NJLRiZIb9CFuBJdPyi+MBoLCMd/LnYygZMYhjzpUCXiHwCy" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=-8\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)+24\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=-8\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)+24\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "Evaluate $$-8\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)+24\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)\\:$$at point $$θ=0:{\\quad}16$$", "result": "=16", "steps": [ { "type": "step", "primary": "Take the point $$θ=0$$ and plug it into $$-8\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)+24\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)$$", "result": "=-8\\left(-2\\sec^{2}\\left(0\\right)+3\\sec^{4}\\left(0\\right)\\right)+24\\left(4\\sec^{4}\\left(0\\right)\\tan^{2}\\left(0\\right)+\\sec^{6}\\left(0\\right)\\right)" }, { "type": "interim", "title": "$$8\\left(-2\\sec^{2}\\left(0\\right)+3\\sec^{4}\\left(0\\right)\\right)=8$$", "input": "8\\left(-2\\sec^{2}\\left(0\\right)+3\\sec^{4}\\left(0\\right)\\right)", "steps": [ { "type": "interim", "title": "$$2\\sec^{2}\\left(0\\right)=2$$", "input": "2\\sec^{2}\\left(0\\right)", "steps": [ { "type": "interim", "title": "$$\\sec^{2}\\left(0\\right)=1$$", "input": "\\sec^{2}\\left(0\\right)", "steps": [ { "type": "interim", "title": "Simplify $$\\sec\\left(0\\right):{\\quad}1$$", "input": "\\sec\\left(0\\right)", "result": "=1^{2}", "steps": [ { "type": "step", "primary": "Use the following trivial identity:$${\\quad}\\sec\\left(0\\right)=1$$", "secondary": [ "$$\\sec\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sec(x)&x&\\sec(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{2\\sqrt{3}}{3}&\\frac{7π}{6}&-\\frac{2\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&\\sqrt{2}&\\frac{5π}{4}&-\\sqrt{2}\\\\\\hline \\frac{π}{3}&2&\\frac{4π}{3}&-2\\\\\\hline \\frac{π}{2}&\\mathrm{Undefined}&\\frac{3π}{2}&\\mathrm{Undefined}\\\\\\hline \\frac{2π}{3}&-2&\\frac{5π}{3}&2\\\\\\hline \\frac{3π}{4}&-\\sqrt{2}&\\frac{7π}{4}&\\sqrt{2}\\\\\\hline \\frac{5π}{6}&-\\frac{2\\sqrt{3}}{3}&\\frac{11π}{6}&\\frac{2\\sqrt{3}}{3}\\\\\\hline \\end{array}$$" ], "result": "=1" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7y1xaZeQTSzp609voOcYk6VXTSum/z5kLpMzXS1UJIexiYFOaxxQqg3o8CNKkTm2qW+Sc57RV820557NJqaC71BfVjTo90KMWspnEdE4CKjE=" } }, { "type": "step", "result": "=2\\cdot\\:1" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=2" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7WiIIsb80YMnC/PcLPmbAry061ljBSPJeENOw2efoSWt8rNweSBKaNqhWM5iGGSVMdOMUzZRHPIlp1R8T2pkp20fQYskTwd8nVLo5risF1/I=" } }, { "type": "interim", "title": "$$3\\sec^{4}\\left(0\\right)=3$$", "input": "3\\sec^{4}\\left(0\\right)", "steps": [ { "type": "interim", "title": "$$\\sec^{4}\\left(0\\right)=1$$", "input": "\\sec^{4}\\left(0\\right)", "steps": [ { "type": "interim", "title": "Simplify $$\\sec\\left(0\\right):{\\quad}1$$", "input": "\\sec\\left(0\\right)", "result": "=1^{4}", "steps": [ { "type": "step", "primary": "Use the following trivial identity:$${\\quad}\\sec\\left(0\\right)=1$$", "secondary": [ "$$\\sec\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sec(x)&x&\\sec(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{2\\sqrt{3}}{3}&\\frac{7π}{6}&-\\frac{2\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&\\sqrt{2}&\\frac{5π}{4}&-\\sqrt{2}\\\\\\hline \\frac{π}{3}&2&\\frac{4π}{3}&-2\\\\\\hline \\frac{π}{2}&\\mathrm{Undefined}&\\frac{3π}{2}&\\mathrm{Undefined}\\\\\\hline \\frac{2π}{3}&-2&\\frac{5π}{3}&2\\\\\\hline \\frac{3π}{4}&-\\sqrt{2}&\\frac{7π}{4}&\\sqrt{2}\\\\\\hline \\frac{5π}{6}&-\\frac{2\\sqrt{3}}{3}&\\frac{11π}{6}&\\frac{2\\sqrt{3}}{3}\\\\\\hline \\end{array}$$" ], "result": "=1" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7TnbPBMsEd78kmmwfYDHqNVXTSum/z5kLpMzXS1UJIexiYFOaxxQqg3o8CNKkTm2qW+Sc57RV820557NJqaC71Ka6Ma3rg33eDIhfX/iuVnM=" } }, { "type": "step", "result": "=3\\cdot\\:1" }, { "type": "step", "primary": "Multiply the numbers: $$3\\cdot\\:1=3$$", "result": "=3" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/DCWessgznXvbIt+KUH6Sy061ljBSPJeENOw2efoSWuWTYGorfWzSEgF1o3crWNtABnhaHvVHSBGAVNgbL75+DAm81Br6rrrU7LVgwPHPmM=" } }, { "type": "step", "result": "=8\\left(3-2\\right)" }, { "type": "step", "primary": "Add/Subtract the numbers: $$-2+3=1$$", "result": "=8\\cdot\\:1" }, { "type": "step", "primary": "Multiply the numbers: $$8\\cdot\\:1=8$$", "result": "=8" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ud5AV3Wb+Jl83O21Lr9iGTOWn8zcWOZXWMkYneaKW/PTLx8mOdHYVzxX643JqKFIF4hPIW9whzh1vNltYmv2zI08PkFfwa+ZiJNMY6gHh8PV06frS5K/+G+6QAS0YeGx0cBjHOAJBd4mS5BoT5dtZ7CI2sSeA74029n2yo277ZU=" } }, { "type": "interim", "title": "$$24\\left(4\\sec^{4}\\left(0\\right)\\tan^{2}\\left(0\\right)+\\sec^{6}\\left(0\\right)\\right)=24$$", "input": "24\\left(4\\sec^{4}\\left(0\\right)\\tan^{2}\\left(0\\right)+\\sec^{6}\\left(0\\right)\\right)", "steps": [ { "type": "interim", "title": "$$4\\sec^{4}\\left(0\\right)\\tan^{2}\\left(0\\right)=0$$", "input": "4\\sec^{4}\\left(0\\right)\\tan^{2}\\left(0\\right)", "steps": [ { "type": "interim", "title": "$$\\sec^{4}\\left(0\\right)=1$$", "input": "\\sec^{4}\\left(0\\right)", "steps": [ { "type": "interim", "title": "Simplify $$\\sec\\left(0\\right):{\\quad}1$$", "input": "\\sec\\left(0\\right)", "result": "=1^{4}", "steps": [ { "type": "step", "primary": "Use the following trivial identity:$${\\quad}\\sec\\left(0\\right)=1$$", "secondary": [ "$$\\sec\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sec(x)&x&\\sec(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{2\\sqrt{3}}{3}&\\frac{7π}{6}&-\\frac{2\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&\\sqrt{2}&\\frac{5π}{4}&-\\sqrt{2}\\\\\\hline \\frac{π}{3}&2&\\frac{4π}{3}&-2\\\\\\hline \\frac{π}{2}&\\mathrm{Undefined}&\\frac{3π}{2}&\\mathrm{Undefined}\\\\\\hline \\frac{2π}{3}&-2&\\frac{5π}{3}&2\\\\\\hline \\frac{3π}{4}&-\\sqrt{2}&\\frac{7π}{4}&\\sqrt{2}\\\\\\hline \\frac{5π}{6}&-\\frac{2\\sqrt{3}}{3}&\\frac{11π}{6}&\\frac{2\\sqrt{3}}{3}\\\\\\hline \\end{array}$$" ], "result": "=1" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7TnbPBMsEd78kmmwfYDHqNVXTSum/z5kLpMzXS1UJIexiYFOaxxQqg3o8CNKkTm2qW+Sc57RV820557NJqaC71Ka6Ma3rg33eDIhfX/iuVnM=" } }, { "type": "step", "result": "=4\\cdot\\:1\\cdot\\:\\tan^{2}\\left(0\\right)" }, { "type": "interim", "title": "$$\\tan^{2}\\left(0\\right)=0$$", "input": "\\tan^{2}\\left(0\\right)", "steps": [ { "type": "interim", "title": "Simplify $$\\tan\\left(0\\right):{\\quad}0$$", "input": "\\tan\\left(0\\right)", "result": "=0^{2}", "steps": [ { "type": "step", "primary": "Use the following trivial identity:$${\\quad}\\tan\\left(0\\right)=0$$", "secondary": [ "$$\\tan\\left(x\\right)$$ periodicity table with $$πn$$ cycle:<br/>$$\\begin{array}{|c|c|}\\hline x&\\tan(x)\\\\\\hline 0&0\\\\\\hline \\frac{π}{6}&\\frac{\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&1\\\\\\hline \\frac{π}{3}&\\sqrt{3}\\\\\\hline \\frac{π}{2}&\\pm\\infty\\\\\\hline \\frac{2π}{3}&-\\sqrt{3}\\\\\\hline \\frac{3π}{4}&-1\\\\\\hline \\frac{5π}{6}&-\\frac{\\sqrt{3}}{3}\\\\\\hline &\\\\\\hline \\end{array}$$" ], "result": "=0" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Apply rule $$0^{a}=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7EFsToMX1NPjPuXVSALucCFXTSum/z5kLpMzXS1UJIeyqjfZqhkrOoAz8sRb7wXZWPGHkI0MFUiTEwg/g9Re5WBSGyzDy0veA23V3wI6HDZ0=" } }, { "type": "step", "result": "=4\\cdot\\:1\\cdot\\:0" }, { "type": "step", "primary": "Apply rule $$0\\cdot\\:a=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s78lx/rXvIfCfIw4X8Nv9JwGpb6Zt5uD6Xe9ICIhSFAqbdd47a0hQ8flDbGsI5To1dTbAOxT8wOTlsw5gGf+Hdr8r4WVs4CtopdU5PWLTw1eRWrxRA8eaaGXYlDOSEM378ENKOm3B8+xM3yGSItlc9gA==" } }, { "type": "interim", "title": "$$\\sec^{6}\\left(0\\right)=1$$", "input": "\\sec^{6}\\left(0\\right)", "steps": [ { "type": "interim", "title": "Simplify $$\\sec\\left(0\\right):{\\quad}1$$", "input": "\\sec\\left(0\\right)", "result": "=1^{6}", "steps": [ { "type": "step", "primary": "Use the following trivial identity:$${\\quad}\\sec\\left(0\\right)=1$$", "secondary": [ "$$\\sec\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sec(x)&x&\\sec(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{2\\sqrt{3}}{3}&\\frac{7π}{6}&-\\frac{2\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&\\sqrt{2}&\\frac{5π}{4}&-\\sqrt{2}\\\\\\hline \\frac{π}{3}&2&\\frac{4π}{3}&-2\\\\\\hline \\frac{π}{2}&\\mathrm{Undefined}&\\frac{3π}{2}&\\mathrm{Undefined}\\\\\\hline \\frac{2π}{3}&-2&\\frac{5π}{3}&2\\\\\\hline \\frac{3π}{4}&-\\sqrt{2}&\\frac{7π}{4}&\\sqrt{2}\\\\\\hline \\frac{5π}{6}&-\\frac{2\\sqrt{3}}{3}&\\frac{11π}{6}&\\frac{2\\sqrt{3}}{3}\\\\\\hline \\end{array}$$" ], "result": "=1" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7WcSYqbVPn/qsbXXyqHdMbFXTSum/z5kLpMzXS1UJIexiYFOaxxQqg3o8CNKkTm2qW+Sc57RV820557NJqaC71Idwe88IvH72vjqcga7gyBg=" } }, { "type": "step", "result": "=24\\left(0+1\\right)" }, { "type": "step", "primary": "Add the numbers: $$0+1=1$$", "result": "=24\\cdot\\:1" }, { "type": "step", "primary": "Multiply the numbers: $$24\\cdot\\:1=24$$", "result": "=24" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7jhDHfABKmSqcJJ1fAFAsGP6k/0Ej05jaZlvKuZ7BchqkNRU526e9DUedYVjkPVRIICf2WQN9mSJxQaQ7cQX4ihqhOePlsg8wUpFvRh9YINqX4WEM7dHJjGE4MiySYuBGWfKjICzStUBkrvQwB+qiDavP5dLtToEw2Zpkikf00/b8OMMqx5/mls5A/L389+Uv" } }, { "type": "step", "result": "=-8+24" }, { "type": "step", "primary": "Simplify" }, { "type": "step", "result": "=16" } ], "meta": { "solvingClass": "Solver", "interimType": "Laplace Eval Function At Point Specific 3Eq" } }, { "type": "step", "result": "16" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\frac{d^{6}}{dθ^{6}}\\left(\\tan\\left(θ\\right)\\right)\\left(0\\right){\\quad:\\quad}0$$", "input": "\\frac{d^{6}}{dθ^{6}}\\left(\\tan\\left(θ\\right)\\right)\\left(0\\right)", "steps": [ { "type": "interim", "title": "$$\\frac{d^{6}}{dθ^{6}}\\left(\\tan\\left(θ\\right)\\right)=-8\\left(-4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+24\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)$$", "input": "\\frac{d^{6}}{dθ^{6}}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "result": "=\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYs0Vsx3c0nt53Utu5DgHTe3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK00HV/bLwwglXz/2YNx2i4JHggSrmCoGjSdud4CyOvVOgdQN0k+N3NHNAkEIoLrhB631k80nn+ODwI4ID0kCF5+z7zRz81SqRD02HEUE4Wqh" } }, { "type": "step", "result": "=\\frac{d^{5}}{dθ^{5}}\\left(\\sec^{2}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{2},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{2}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{2}\\right)=2u$$", "input": "\\frac{d}{du}\\left(u^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2u^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2u", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYr+VZUwBnLdzbS6DZQ+f4s+k3hxk9aCfAWodBRxXgUexMchyqTAJWrzJaDbnNcFsJUeCBKuYKgaNJ253gLI69U79qbCA2QqVmvm3jGRXZ2ppvbGT4j1utMEkCDH25m/vlQ==" } }, { "type": "step", "result": "=2u\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgCSXIThcWH58oYoMPt18lzFqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidZClirlWNFPnHR1rvdyoFLI7IC+oqnwb1T6nPLpiw1eysw2NXPd8jZnr3fcPpqc7ejeh7+jKEzLb7VNCEMF3Z/Maswa5oIagKQFOfrnKm5+Gbb/PxX3awut5KxRxyZf8A=" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{1+1}\\left(θ\\right)$$" ], "result": "=2\\sec^{1+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7aY4rYekMZhUfP6sZ6mfqd6Lsz0lt1XLEr88Kc/cjJo5V00rpv8+ZC6TM10tVCSHspqrCcTwb/I0f8OwMjuU6eBAUa+AHsHd5OGpMiuFITjOuqvW3hAHH635V/vEEYIatHimBRYRqHSWeJkuUPhfTC5LQ0B1JNgsb5SXzbS+SCnAfGB9H25Zc+Ua1z7Kw8kgb4gBJl4WMO1rA0a30/bUYlg==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{d^{4}}{dθ^{4}}\\left(2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\right)=-4\\sec^{2}\\left(θ\\right)+6\\sec^{4}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=2\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the Product Rule: $$\\left(f{\\cdot}g\\right)'=f'{\\cdot}g+f{\\cdot}g'$$", "secondary": [ "$$f=\\sec^{2}\\left(θ\\right),\\:g=\\tan\\left(θ\\right)$$" ], "result": "=2\\left(\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)\\tan\\left(θ\\right)+\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)\\sec^{2}\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Product%20Rule", "practiceTopic": "Product Rule" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{2},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{2}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{2}\\right)=2u$$", "input": "\\frac{d}{du}\\left(u^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2u^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2u", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYr+VZUwBnLdzbS6DZQ+f4s+k3hxk9aCfAWodBRxXgUexMchyqTAJWrzJaDbnNcFsJUeCBKuYKgaNJ253gLI69U79qbCA2QqVmvm3jGRXZ2ppvbGT4j1utMEkCDH25m/vlQ==" } }, { "type": "step", "result": "=2u\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgCSXIThcWH58oYoMPt18lzFqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidZClirlWNFPnHR1rvdyoFLI7IC+oqnwb1T6nPLpiw1eysw2NXPd8jZnr3fcPpqc7ejeh7+jKEzLb7VNCEMF3Z/Maswa5oIagKQFOfrnKm5+Gbb/PxX3awut5KxRxyZf8A=" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{1+1}\\left(θ\\right)$$" ], "result": "=2\\sec^{1+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7aY4rYekMZhUfP6sZ6mfqd6Lsz0lt1XLEr88Kc/cjJo5V00rpv8+ZC6TM10tVCSHspqrCcTwb/I0f8OwMjuU6eBAUa+AHsHd5OGpMiuFITjOuqvW3hAHH635V/vEEYIatHimBRYRqHSWeJkuUPhfTC5LQ0B1JNgsb5SXzbS+SCnAfGB9H25Zc+Ua1z7Kw8kgb4gBJl4WMO1rA0a30/bUYlg==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "result": "=\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYs0Vsx3c0nt53Utu5DgHTe3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK00HV/bLwwglXz/2YNx2i4JHggSrmCoGjSdud4CyOvVOgdQN0k+N3NHNAkEIoLrhB631k80nn+ODwI4ID0kCF5+z7zRz81SqRD02HEUE4Wqh" } }, { "type": "step", "result": "=2\\left(2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Simplify $$2\\left(2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\right):{\\quad}-4\\sec^{2}\\left(θ\\right)+6\\sec^{4}\\left(θ\\right)$$", "input": "2\\left(2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\right)", "result": "=-4\\sec^{2}\\left(θ\\right)+6\\sec^{4}\\left(θ\\right)", "steps": [ { "type": "interim", "title": "$$2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)=2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)$$", "input": "2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\tan\\left(θ\\right)\\tan\\left(θ\\right)=\\:\\tan^{1+1}\\left(θ\\right)$$" ], "result": "=2\\sec^{2}\\left(θ\\right)\\tan^{1+1}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Md1SJKHYLxhOYRRbsLTu+axUZgt6FTmFSHnRsotkqemLeJKmeakO1+Q3aSIldG41s7BwSVdKEYSqn+4fKAQoGk0xmFIl1KTO5aAkhaHvKzBALhShbMGZIZsZWTO721HsWI4dG63F/aSz/PFDvu5FnO3sngYNzcydIedGQjjtM8sBMUhkLMqDD6KHvgZAgKMdRQzIIgXxnWeDhDTk2Lfum8RPUSGriaFRn0NZXHZrPQw=" } }, { "type": "interim", "title": "$$\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)=\\sec^{4}\\left(θ\\right)$$", "input": "\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)=\\:\\sec^{2+2}\\left(θ\\right)$$" ], "result": "=\\sec^{2+2}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+2=4$$", "result": "=\\sec^{4}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7AtrsReczPdUqIdn5/Ev5mFtNfVJqszzpMwg6spHO8F8gJ/ZZA32ZInFBpDtxBfiKn3Jyj2VJgBojKQHjINpkkETYvnjqwhFu6aWmDOYpAVMO/U6qyRZonn1WPQ/21pz3cKXVLLQXMot2gtnZH7qY5zkbLioo7YVO0SRnDAO/7Uokt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "=2\\left(\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Expand $$\\left(\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)\\cdot\\:2:{\\quad}2\\sec^{4}\\left(θ\\right)+4\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)$$", "input": "\\left(\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)\\cdot\\:2", "result": "=2\\sec^{4}\\left(θ\\right)+4\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)", "steps": [ { "type": "step", "result": "=2\\left(\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the distributive law: $$a\\left(b+c\\right)=ab+ac$$", "secondary": [ "$$a=2,\\:b=\\sec^{4}\\left(θ\\right),\\:c=2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)$$" ], "result": "=2\\sec^{4}\\left(θ\\right)+2\\cdot\\:2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=2\\sec^{4}\\left(θ\\right)+4\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7cTiUO1fOFax2wXHZewjhxEUAdh/O2PLAxzMjSpKce9YL4tWjzbYCZhBh/Hc2rUXacMKakGXZAry+N/hvJOodjXCQoYlYQ8U+Tfyx0kyzI8iF3F2TkMs7mKjIsUPAR/Vd82dH/qvnCC7/oZy4HZlX2aHh6acbPp8G70jj/skoFx7vbBmbuQNTF0TphKZ8Ruvacu+LCkBvFQsw7xBL2Ys2TUYb1899n2cPVSJTSzzdZzp+38s+uQz1W6FLDk3KkIYWhJuRFkG6Ai089lDjeB1lMbCI2sSeA74029n2yo277ZU=" } }, { "type": "interim", "title": "Rewrite using trig identities", "input": "2\\sec^{4}\\left(θ\\right)+4\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)", "result": "=-4\\sec^{2}\\left(θ\\right)+6\\sec^{4}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Use the Pythagorean identity: $$\\tan^{2}\\left(x\\right)+1=\\sec^{2}\\left(x\\right)$$", "secondary": [ "$$\\tan^{2}\\left(x\\right)=\\sec^{2}\\left(x\\right)-1$$" ], "result": "=2\\sec^{4}\\left(θ\\right)+4\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right)" }, { "type": "interim", "title": "Simplify $$2\\sec^{4}\\left(θ\\right)+4\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right):{\\quad}6\\sec^{4}\\left(θ\\right)-4\\sec^{2}\\left(θ\\right)$$", "input": "2\\sec^{4}\\left(θ\\right)+4\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right)", "result": "=6\\sec^{4}\\left(θ\\right)-4\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "interim", "title": "Expand $$4\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right):{\\quad}4\\sec^{4}\\left(θ\\right)-4\\sec^{2}\\left(θ\\right)$$", "input": "4\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right)", "result": "=2\\sec^{4}\\left(θ\\right)+4\\sec^{4}\\left(θ\\right)-4\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b-c\\right)=ab-ac$$", "secondary": [ "$$a=4\\sec^{2}\\left(θ\\right),\\:b=\\sec^{2}\\left(θ\\right),\\:c=1$$" ], "result": "=4\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)-4\\sec^{2}\\left(θ\\right)\\cdot\\:1", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "result": "=4\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)-4\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$4\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)-4\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right):{\\quad}4\\sec^{4}\\left(θ\\right)-4\\sec^{2}\\left(θ\\right)$$", "input": "4\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)-4\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right)", "result": "=4\\sec^{4}\\left(θ\\right)-4\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "interim", "title": "$$4\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)=4\\sec^{4}\\left(θ\\right)$$", "input": "4\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)=\\:\\sec^{2+2}\\left(θ\\right)$$" ], "result": "=4\\sec^{2+2}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+2=4$$", "result": "=4\\sec^{4}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vjDr4p6TvcgiSLV1uDo4U9JYPDPDesonRI520G8zYRPehkKrn0era9rz8TlL+x/vMHVZf8xiryCPkbUIYXhiIHRfFgXB+vlRb/AHCJklpF9e7YF3TFAJSjcLX2VvhbubqX/cSuhK5Ty2xqd/IdNr6zlZ8CVoKZl2BAoDdAtcFNxU2fg2Eo8khoQfnK/P3ock" } }, { "type": "interim", "title": "$$4\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right)=4\\sec^{2}\\left(θ\\right)$$", "input": "4\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:1=4$$", "result": "=4\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7xF9rlQkuAOrRzKTf6eQs7fgkIqVBCzsTrV5i/ECNCw8tOtZYwUjyXhDTsNnn6Elr/BCsllMMEyc8UGKoUwuYxu6vfsn/ogOG+fHMfNQh9JeZfoRPsbCjih+xtMTOMGl4tzgLeG7CLgj8aHpc44iME/LWpEthEIQrAjNHY4qDKt/ET1Ehq4mhUZ9DWVx2az0M" } }, { "type": "step", "result": "=4\\sec^{4}\\left(θ\\right)-4\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vjDr4p6TvcgiSLV1uDo4U/jbMbEoNNbIR/7Q84OynCfTLx8mOdHYVzxX643JqKFIvTZkYD4gUeSoFrhGGhxq98kwYX4VRPb1geKkn+NU7mcE3SiHEhDlxIXJDVY+PPdvgQUxJPyUNnGfVirkcwpVO84fwxlvni/T8DIWnD0mw9U6FFV/0Hm6Y1hI7VaPzzKhkSoZP7CyArJlOo0FRny79Q==" } }, { "type": "step", "primary": "Add similar elements: $$2\\sec^{4}\\left(θ\\right)+4\\sec^{4}\\left(θ\\right)=6\\sec^{4}\\left(θ\\right)$$", "result": "=6\\sec^{4}\\left(θ\\right)-4\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7f48emNsIl4MI4hQZuZje4penFTHzh6X+DX3uvPdqYy7umM/GfBKdAtyuCOAeIKZmAJYpRu9XpYrd8NSAW2DdD0c/CXknyRKFbCevkP3vkuVyZXAw9UlCE0mEZtdsFPrV7q9+yf+iA4b58cx81CH0l/C30sSftAIFS6Qkpy19Ikp0x8irWzKvvppKMUvqTrnrUObZozYPTEmgGDR5V3grHDoUVX/QebpjWEjtVo/PMqGRKhk/sLICsmU6jQVGfLv1" } } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7xqDNX42OyJjPevbF+7IBng7708BkbVE1gb8/0kMEgzvXUIQv0aG2ncbDgSWX/DWycNZx/2EDub2dSsLw3NGg9xLC2T8NFZvTA+pff5AdXPdX9lRN4mlYq/zI0dgxEXGdPvdnhi0s3lZ+DJ7rEiTeqenxKTmAB1ahPFE542Dd/TeutOLi0kB5xHysaBsHIgntIaTy8gmKy5PY3el+RbUmlkmFP65WPUHCIGTpzT/EAccetg60+ZUlqP5jB+ReldUA5AMRxXWt+t9va0qZyagxyQ==" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7xI7dknI212svHgZHHIcoRvkdSragnZ5K8+WMRM5qjJ6ETkUamdUz59t0aHpV3uN0rE0Ib0rKQ2oM97vl/gDbowCWKUbvV6WK3fDUgFtg3Q+MfZ1BFRhhGnQf9q28g9nK/SzNF20Xrftd883pI0RALQ5+VkRmueXOP55Piqx5MyWBBTEk/JQ2cZ9WKuRzClU7m4RppR4RRNJR6+bqv/lHExUnakC9Y1jhr4uI97E9yvalDk6K3L00/zgNSSliALTtnpy3tAot7+5PIP9fBKZw2Llxun299aFzbjCyRFblUKE=" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{d^{3}}{dθ^{3}}\\left(-4\\sec^{2}\\left(θ\\right)+6\\sec^{4}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(-4\\sec^{2}\\left(θ\\right)+6\\sec^{4}\\left(θ\\right)\\right)=-8\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+24\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(-4\\sec^{2}\\left(θ\\right)+6\\sec^{4}\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=-\\frac{d}{dθ}\\left(4\\sec^{2}\\left(θ\\right)\\right)+\\frac{d}{dθ}\\left(6\\sec^{4}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(4\\sec^{2}\\left(θ\\right)\\right)=8\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(4\\sec^{2}\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=4\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Apply the chain rule:$${\\quad}2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{2},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{2}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{2}\\right)=2u$$", "input": "\\frac{d}{du}\\left(u^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2u^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2u", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYr+VZUwBnLdzbS6DZQ+f4s+k3hxk9aCfAWodBRxXgUexMchyqTAJWrzJaDbnNcFsJUeCBKuYKgaNJ253gLI69U79qbCA2QqVmvm3jGRXZ2ppvbGT4j1utMEkCDH25m/vlQ==" } }, { "type": "step", "result": "=2u\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgCSXIThcWH58oYoMPt18lzFqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidZClirlWNFPnHR1rvdyoFLI7IC+oqnwb1T6nPLpiw1eysw2NXPd8jZnr3fcPpqc7ejeh7+jKEzLb7VNCEMF3Z/Maswa5oIagKQFOfrnKm5+Gbb/PxX3awut5KxRxyZf8A=" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=4\\cdot\\:2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$4\\cdot\\:2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}8\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "4\\cdot\\:2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=8\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:2=8$$", "result": "=8\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{1+1}\\left(θ\\right)$$" ], "result": "=8\\sec^{1+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=8\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7NqBYpvGKZC5lMRQos2qph5K/zREn/U8meKQ3TvRjTQz8cZMsH3IX6lxp+ANHGbuCq47vuWedXv2WUg94ER8IwTfgzf+5OuhkULh+nunjrqae0Rv083m3K3ie3DVArOu08LfSxJ+0AgVLpCSnLX0iSqXiJoCK22T5YMsRbcSz8DsyFZaieHAvmtaw5r6RKguDHxgfR9uWXPlGtc+ysPJIG+IASZeFjDtawNGt9P21GJY=" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(6\\sec^{4}\\left(θ\\right)\\right)=24\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(6\\sec^{4}\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=6\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Apply the chain rule:$${\\quad}4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{4},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{4}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{4}\\right)=4u^{3}$$", "input": "\\frac{d}{du}\\left(u^{4}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=4u^{4-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=4u^{3}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYl3oEe4EC6n4B8Xgz0cKnkqk3hxk9aCfAWodBRxXgUexuzo0Fl6VhneutEteidn+Kv8//6/nV5O4fb8Xgwi7mapNFxUvwBeni+JEIFAdbegzYEOcQgyW4n51cR7AeMAm7Q==" } }, { "type": "step", "result": "=4u^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgA2CO5cm49xNEqFV0DMaZJRqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidVduWEnoDYDg06IbaTsp9TFtqiuNhFTvmF8ZCDqWZNoubfnOah8gAvLMKOeNMctC5tVYZ/3r82LxPimKUFXjQFStvD8GKDNJJN8qiu0XRcVZAYAvbRKqkRVDaDBHkSBtDwYHkfmHHTze3JXk3LjiJp" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=6\\cdot\\:4\\left(\\sec\\left(θ\\right)\\right)^{3}\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$6\\cdot\\:4\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}24\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "6\\cdot\\:4\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=24\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$6\\cdot\\:4=24$$", "result": "=24\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{3+1}\\left(θ\\right)$$" ], "result": "=24\\sec^{3+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$3+1=4$$", "result": "=24\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l3FkS70/4BFsebxcaZbHnTNSoP5XQt+n/5y371e8QHJROlTYN926lBwmiJdD5HUSzRqDxPUzBN6vjj5oJL9kUIdu2MPseykkfgn/Trt10KGp9ej8WEAztP3UNEC9n7jjZEt3ZXAiqUE0HIXrrrezJHZKl2mV4AlztiApTAFigz07AW6Jz0sXkPNp6b6xpHMqM1Kg/ldC36f/nLfvV7xActytFlzCV8oq55bHLwXbb0c=" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=-8\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+24\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{d^{2}}{dθ^{2}}\\left(-8\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+24\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(-8\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+24\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)=-8\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)+24\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(-8\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+24\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=-\\frac{d}{dθ}\\left(8\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+\\frac{d}{dθ}\\left(24\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(8\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\right)=8\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(8\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=8\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the Product Rule: $$\\left(f{\\cdot}g\\right)'=f'{\\cdot}g+f{\\cdot}g'$$", "secondary": [ "$$f=\\sec^{2}\\left(θ\\right),\\:g=\\tan\\left(θ\\right)$$" ], "result": "=8\\left(\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)\\tan\\left(θ\\right)+\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)\\sec^{2}\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Product%20Rule", "practiceTopic": "Product Rule" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{2},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{2}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{2}\\right)=2u$$", "input": "\\frac{d}{du}\\left(u^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2u^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2u", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYr+VZUwBnLdzbS6DZQ+f4s+k3hxk9aCfAWodBRxXgUexMchyqTAJWrzJaDbnNcFsJUeCBKuYKgaNJ253gLI69U79qbCA2QqVmvm3jGRXZ2ppvbGT4j1utMEkCDH25m/vlQ==" } }, { "type": "step", "result": "=2u\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgCSXIThcWH58oYoMPt18lzFqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidZClirlWNFPnHR1rvdyoFLI7IC+oqnwb1T6nPLpiw1eysw2NXPd8jZnr3fcPpqc7ejeh7+jKEzLb7VNCEMF3Z/Maswa5oIagKQFOfrnKm5+Gbb/PxX3awut5KxRxyZf8A=" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{1+1}\\left(θ\\right)$$" ], "result": "=2\\sec^{1+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7aY4rYekMZhUfP6sZ6mfqd6Lsz0lt1XLEr88Kc/cjJo5V00rpv8+ZC6TM10tVCSHspqrCcTwb/I0f8OwMjuU6eBAUa+AHsHd5OGpMiuFITjOuqvW3hAHH635V/vEEYIatHimBRYRqHSWeJkuUPhfTC5LQ0B1JNgsb5SXzbS+SCnAfGB9H25Zc+Ua1z7Kw8kgb4gBJl4WMO1rA0a30/bUYlg==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "result": "=\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYs0Vsx3c0nt53Utu5DgHTe3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK00HV/bLwwglXz/2YNx2i4JHggSrmCoGjSdud4CyOvVOgdQN0k+N3NHNAkEIoLrhB631k80nn+ODwI4ID0kCF5+z7zRz81SqRD02HEUE4Wqh" } }, { "type": "step", "result": "=8\\left(2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Simplify $$8\\left(2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\right):{\\quad}8\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)$$", "input": "8\\left(2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\right)", "result": "=8\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "$$2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)=2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)$$", "input": "2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\tan\\left(θ\\right)\\tan\\left(θ\\right)=\\:\\tan^{1+1}\\left(θ\\right)$$" ], "result": "=2\\sec^{2}\\left(θ\\right)\\tan^{1+1}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Md1SJKHYLxhOYRRbsLTu+axUZgt6FTmFSHnRsotkqemLeJKmeakO1+Q3aSIldG41s7BwSVdKEYSqn+4fKAQoGk0xmFIl1KTO5aAkhaHvKzBALhShbMGZIZsZWTO721HsWI4dG63F/aSz/PFDvu5FnO3sngYNzcydIedGQjjtM8sBMUhkLMqDD6KHvgZAgKMdRQzIIgXxnWeDhDTk2Lfum8RPUSGriaFRn0NZXHZrPQw=" } }, { "type": "interim", "title": "$$\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)=\\sec^{4}\\left(θ\\right)$$", "input": "\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)=\\:\\sec^{2+2}\\left(θ\\right)$$" ], "result": "=\\sec^{2+2}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+2=4$$", "result": "=\\sec^{4}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7AtrsReczPdUqIdn5/Ev5mFtNfVJqszzpMwg6spHO8F8gJ/ZZA32ZInFBpDtxBfiKn3Jyj2VJgBojKQHjINpkkETYvnjqwhFu6aWmDOYpAVMO/U6qyRZonn1WPQ/21pz3cKXVLLQXMot2gtnZH7qY5zkbLioo7YVO0SRnDAO/7Uokt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "=8\\left(\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Simplify $$2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{4}\\left(θ\\right):{\\quad}3\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)$$", "input": "2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{4}\\left(θ\\right)", "result": "=8\\left(3\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Rewrite using trig identities", "input": "\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)", "result": "=-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Use the Pythagorean identity: $$\\tan^{2}\\left(x\\right)+1=\\sec^{2}\\left(x\\right)$$", "secondary": [ "$$\\tan^{2}\\left(x\\right)=\\sec^{2}\\left(x\\right)-1$$" ], "result": "=\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right)" }, { "type": "interim", "title": "Simplify $$\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right):{\\quad}3\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)$$", "input": "\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right)", "result": "=3\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "interim", "title": "Expand $$2\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right):{\\quad}2\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)$$", "input": "2\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right)", "result": "=\\sec^{4}\\left(θ\\right)+2\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b-c\\right)=ab-ac$$", "secondary": [ "$$a=2\\sec^{2}\\left(θ\\right),\\:b=\\sec^{2}\\left(θ\\right),\\:c=1$$" ], "result": "=2\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)\\cdot\\:1", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "result": "=2\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)-2\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$2\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)-2\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right):{\\quad}2\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)$$", "input": "2\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)-2\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right)", "result": "=2\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "interim", "title": "$$2\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)=2\\sec^{4}\\left(θ\\right)$$", "input": "2\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)=\\:\\sec^{2+2}\\left(θ\\right)$$" ], "result": "=2\\sec^{2+2}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+2=4$$", "result": "=2\\sec^{4}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Md1SJKHYLxhOYRRbsLTu+dJYPDPDesonRI520G8zYRPehkKrn0era9rz8TlL+x/vyvn14PieOyuPP98CkiPIaXRfFgXB+vlRb/AHCJklpF+qQC22PXYzQL8KzAhJCzszqX/cSuhK5Ty2xqd/IdNr67A2LXxYB8f3C3KpWaLgll1U2fg2Eo8khoQfnK/P3ock" } }, { "type": "interim", "title": "$$2\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right)=2\\sec^{2}\\left(θ\\right)$$", "input": "2\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=2\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ht1jgkupea8+jUgT+M/VFPgkIqVBCzsTrV5i/ECNCw8tOtZYwUjyXhDTsNnn6Elr8U8lPA4O6WXezAN1J2P7Ue6vfsn/ogOG+fHMfNQh9Je7opJQAzFYPpqn/2bLavXutzgLeG7CLgj8aHpc44iMEx91+u64Nzpp2/441mWGcQ/ET1Ehq4mhUZ9DWVx2az0M" } }, { "type": "step", "result": "=2\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Md1SJKHYLxhOYRRbsLTu+fjbMbEoNNbIR/7Q84OynCfTLx8mOdHYVzxX643JqKFI1Jetp9AvoQcY2QSDwFJhz1kUvGSawlk7umQ8kurx5doE3SiHEhDlxIXJDVY+PPdvgQUxJPyUNnGfVirkcwpVO6RMkE3rByM/Y1Dxxfs3xFw6FFV/0Hm6Y1hI7VaPzzKhkSoZP7CyArJlOo0FRny79Q==" } }, { "type": "step", "primary": "Add similar elements: $$\\sec^{4}\\left(θ\\right)+2\\sec^{4}\\left(θ\\right)=3\\sec^{4}\\left(θ\\right)$$", "result": "=3\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ARkhlGZdazrdxNLzhlqePtRiXXTzenxbt00ahL9sDS5L0MjjUe01mdeSwRMoYPSBLTrWWMFI8l4Q07DZ5+hJazaxvejhvTJvqoS105+T9+xc3qH/hVuCay9igw/1IwZBNySLUGdPDJeW3uY1XSs479bA+zX4bD3u3gx65o2NJhP8KqIsxz0lwScBMkZS31XhRCRwG0yFaMyhS3uC5/uUeMvqafR00cq/gMXQBnztqPn0yTzBsNnlyCSr/zrcxdQG" } } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7FkSOkw7Nc+ISjlBZ7XiESO/dTLaj0YrNSw6+/ogyGX8k+jBZ4XxujMMeS3Qvsi6m7aEe0hvT9VtxXei653t2zQi9Vnn4imguIoWn3N1eqjdGIkb3QW2w7y4N83RSktbc/kUupyp7xXxB8YmKEQ5HoPs4SAUa1zTfp2gkDWnxoYVtNtNgAtA4xHxju0w39zFkZEt3ZXAiqUE0HIXrrrezJJi0cdkgnw2HPjRVpcHMxZbIciYVzjaAolEaNkVONtBBRppTNgMFons9bMkTzIHooA==" } } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7JzuoJyZ0ket+0by442o6nfkdSragnZ5K8+WMRM5qjJ6ETkUamdUz59t0aHpV3uN0rE0Ib0rKQ2oM97vl/gDbowCWKUbvV6WK3fDUgFtg3Q+nr1JqgQ+t+tehVxpQusgo+5k7zGGz9wmA2lyQsVWy+j6F2h9EjPuS9/S594y1Nd1FKk3fejFkyiOiq9iG9IkAIuSmJUDtLxRPyJ57Nkq7qjUyCLMvQZ782OaKpAVh7FOsVGYLehU5hUh50bKLZKnpo1d4a+pBCGznOTLxP2sni3enTnyZLeuZs6pBwuQMplUkt3WiGR7ZaCaXvz77bMjS" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(24\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)=24\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(24\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=24\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the Product Rule: $$\\left(f{\\cdot}g\\right)'=f'{\\cdot}g+f{\\cdot}g'$$", "secondary": [ "$$f=\\sec^{4}\\left(θ\\right),\\:g=\\tan\\left(θ\\right)$$" ], "result": "=24\\left(\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)\\tan\\left(θ\\right)+\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)\\sec^{4}\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Product%20Rule", "practiceTopic": "Product Rule" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)=4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{4},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{4}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{4}\\right)=4u^{3}$$", "input": "\\frac{d}{du}\\left(u^{4}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=4u^{4-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=4u^{3}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYl3oEe4EC6n4B8Xgz0cKnkqk3hxk9aCfAWodBRxXgUexuzo0Fl6VhneutEteidn+Kv8//6/nV5O4fb8Xgwi7mapNFxUvwBeni+JEIFAdbegzYEOcQgyW4n51cR7AeMAm7Q==" } }, { "type": "step", "result": "=4u^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgA2CO5cm49xNEqFV0DMaZJRqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidVduWEnoDYDg06IbaTsp9TFtqiuNhFTvmF8ZCDqWZNoubfnOah8gAvLMKOeNMctC5tVYZ/3r82LxPimKUFXjQFStvD8GKDNJJN8qiu0XRcVZAYAvbRKqkRVDaDBHkSBtDwYHkfmHHTze3JXk3LjiJp" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$4\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "4\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{3+1}\\left(θ\\right)$$" ], "result": "=4\\sec^{3+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$3+1=4$$", "result": "=4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7p9w76+/nvHitDrMIupHrrrPmAPxuc/RFHWXkB8A+mnqLeJKmeakO1+Q3aSIldG41PCyA7ixjCdfDYe+yac7KOvOIrkYdKW3Q58EnXaMU+7oslHPSvPf5UfEs5T+IFZ4XHjb2+5NLFZrsH9fcPWg/TXL71sHPAsAQtYZvjpaGSRwboK/Igc1IRC3EGruCoRzV1Zrc/SYRIR8JuxYpD+PJ+Q==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "result": "=\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYs0Vsx3c0nt53Utu5DgHTe3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK00HV/bLwwglXz/2YNx2i4JHggSrmCoGjSdud4CyOvVOgdQN0k+N3NHNAkEIoLrhB631k80nn+ODwI4ID0kCF5+z7zRz81SqRD02HEUE4Wqh" } }, { "type": "step", "result": "=24\\left(4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Simplify $$24\\left(4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right):{\\quad}24\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)$$", "input": "24\\left(4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right)", "result": "=24\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "$$4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)=4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)$$", "input": "4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\tan\\left(θ\\right)\\tan\\left(θ\\right)=\\:\\tan^{1+1}\\left(θ\\right)$$" ], "result": "=4\\sec^{4}\\left(θ\\right)\\tan^{1+1}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7GHerUg4zB8o0EeBO6BKV16xUZgt6FTmFSHnRsotkqemLeJKmeakO1+Q3aSIldG41PCyA7ixjCdfDYe+yac7KOvOIrkYdKW3Q58EnXaMU+7pALhShbMGZIZsZWTO721HsXaBT4KNKuhZZSFsFWnKRStdNJOkRdCtrWB4iMQVmMwBoAWE3NiWlLPrDNnXLOZLK3tKI9zGyi4xCTaccmboyHcRPUSGriaFRn0NZXHZrPQw=" } }, { "type": "interim", "title": "$$\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)=\\sec^{6}\\left(θ\\right)$$", "input": "\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)=\\:\\sec^{2+4}\\left(θ\\right)$$" ], "result": "=\\sec^{2+4}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+4=6$$", "result": "=\\sec^{6}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7AtrsReczPdUqIdn5/Ev5mLBxohAhN47qWD2nlwOHqfkgJ/ZZA32ZInFBpDtxBfiKn3Jyj2VJgBojKQHjINpkkG6JHhxn2McSW7OxLgQBi64O/U6qyRZonn1WPQ/21pz3xMe3sR8ZMoA2Ynlp2W853t3OcQiSW6rZ4zI81UhADI4kt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "=24\\left(\\sec^{6}\\left(θ\\right)+4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7jhDHfABKmSqcJJ1fAFAsGBiMJmjxigW8LortZKvrqEbWPF6oWuXggupsFvXrooIqxMe3sR8ZMoA2Ynlp2W853tMvHyY50dhXPFfrjcmooUg3CXpyb+Iu3cMrCn545PU8uO1NenZJP60hEwBH05kmlvOIrkYdKW3Q58EnXaMU+7r4ZAjrJM4mwvgfYovsxoopgQUxJPyUNnGfVirkcwpVOxKnUVO0TgjLT50Mmk0DzLYh0PvFonSnuZi7Z6ItIhmYhYbzXwn7iok10o2w+NJLRiZIb9CFuBJdPyi+MBoLCMd/LnYygZMYhjzpUCXiHwCy" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=-8\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)+24\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{d}{dθ}\\left(-8\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)+24\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(-8\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)+24\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)\\right)=-8\\left(-4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+24\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(-8\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)+24\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=-\\frac{d}{dθ}\\left(8\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)\\right)+\\frac{d}{dθ}\\left(24\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(8\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)\\right)=8\\left(-4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(8\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=8\\frac{d}{dθ}\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=8\\left(-\\frac{d}{dθ}\\left(2\\sec^{2}\\left(θ\\right)\\right)+\\frac{d}{dθ}\\left(3\\sec^{4}\\left(θ\\right)\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(2\\sec^{2}\\left(θ\\right)\\right)=4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(2\\sec^{2}\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=2\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Apply the chain rule:$${\\quad}2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{2},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{2}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{2}\\right)=2u$$", "input": "\\frac{d}{du}\\left(u^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2u^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2u", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYr+VZUwBnLdzbS6DZQ+f4s+k3hxk9aCfAWodBRxXgUexMchyqTAJWrzJaDbnNcFsJUeCBKuYKgaNJ253gLI69U79qbCA2QqVmvm3jGRXZ2ppvbGT4j1utMEkCDH25m/vlQ==" } }, { "type": "step", "result": "=2u\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgCSXIThcWH58oYoMPt18lzFqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidZClirlWNFPnHR1rvdyoFLI7IC+oqnwb1T6nPLpiw1eysw2NXPd8jZnr3fcPpqc7ejeh7+jKEzLb7VNCEMF3Z/Maswa5oIagKQFOfrnKm5+Gbb/PxX3awut5KxRxyZf8A=" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=2\\cdot\\:2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$2\\cdot\\:2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "2\\cdot\\:2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=4\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{1+1}\\left(θ\\right)$$" ], "result": "=4\\sec^{1+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s71jEHNygbGF2qQyJrTxQFvpK/zREn/U8meKQ3TvRjTQz8cZMsH3IX6lxp+ANHGbuCq47vuWedXv2WUg94ER8IwfulLd9uW7xFGDkqRxB3vSSe0Rv083m3K3ie3DVArOu08LfSxJ+0AgVLpCSnLX0iSnTHyKtbMq++mkoxS+pOuesyFZaieHAvmtaw5r6RKguDHxgfR9uWXPlGtc+ysPJIG+IASZeFjDtawNGt9P21GJY=" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(3\\sec^{4}\\left(θ\\right)\\right)=12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(3\\sec^{4}\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=3\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Apply the chain rule:$${\\quad}4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{4},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{4}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{4}\\right)=4u^{3}$$", "input": "\\frac{d}{du}\\left(u^{4}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=4u^{4-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=4u^{3}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYl3oEe4EC6n4B8Xgz0cKnkqk3hxk9aCfAWodBRxXgUexuzo0Fl6VhneutEteidn+Kv8//6/nV5O4fb8Xgwi7mapNFxUvwBeni+JEIFAdbegzYEOcQgyW4n51cR7AeMAm7Q==" } }, { "type": "step", "result": "=4u^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgA2CO5cm49xNEqFV0DMaZJRqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidVduWEnoDYDg06IbaTsp9TFtqiuNhFTvmF8ZCDqWZNoubfnOah8gAvLMKOeNMctC5tVYZ/3r82LxPimKUFXjQFStvD8GKDNJJN8qiu0XRcVZAYAvbRKqkRVDaDBHkSBtDwYHkfmHHTze3JXk3LjiJp" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=3\\cdot\\:4\\left(\\sec\\left(θ\\right)\\right)^{3}\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$3\\cdot\\:4\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "3\\cdot\\:4\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$3\\cdot\\:4=12$$", "result": "=12\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{3+1}\\left(θ\\right)$$" ], "result": "=12\\sec^{3+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$3+1=4$$", "result": "=12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7jFGb0q9vd+kdWvVYeHJHbDNSoP5XQt+n/5y371e8QHJROlTYN926lBwmiJdD5HUSzRqDxPUzBN6vjj5oJL9kUFPzsE52Mw7P00gxEE61byKp9ej8WEAztP3UNEC9n7jjZEt3ZXAiqUE0HIXrrrezJHZKl2mV4AlztiApTAFigz2gNkWvAT/c3O3uDO/qxcKwM1Kg/ldC36f/nLfvV7xActytFlzCV8oq55bHLwXbb0c=" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=8\\left(-4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(24\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)\\right)=24\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(24\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=24\\frac{d}{dθ}\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=24\\left(\\frac{d}{dθ}\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)+\\frac{d}{dθ}\\left(\\sec^{6}\\left(θ\\right)\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)=4\\left(4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=4\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the Product Rule: $$\\left(f{\\cdot}g\\right)'=f'{\\cdot}g+f{\\cdot}g'$$", "secondary": [ "$$f=\\sec^{4}\\left(θ\\right),\\:g=\\tan^{2}\\left(θ\\right)$$" ], "result": "=4\\left(\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)\\tan^{2}\\left(θ\\right)+\\frac{d}{dθ}\\left(\\tan^{2}\\left(θ\\right)\\right)\\sec^{4}\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Product%20Rule", "practiceTopic": "Product Rule" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)=4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{4},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{4}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{4}\\right)=4u^{3}$$", "input": "\\frac{d}{du}\\left(u^{4}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=4u^{4-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=4u^{3}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYl3oEe4EC6n4B8Xgz0cKnkqk3hxk9aCfAWodBRxXgUexuzo0Fl6VhneutEteidn+Kv8//6/nV5O4fb8Xgwi7mapNFxUvwBeni+JEIFAdbegzYEOcQgyW4n51cR7AeMAm7Q==" } }, { "type": "step", "result": "=4u^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgA2CO5cm49xNEqFV0DMaZJRqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidVduWEnoDYDg06IbaTsp9TFtqiuNhFTvmF8ZCDqWZNoubfnOah8gAvLMKOeNMctC5tVYZ/3r82LxPimKUFXjQFStvD8GKDNJJN8qiu0XRcVZAYAvbRKqkRVDaDBHkSBtDwYHkfmHHTze3JXk3LjiJp" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$4\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "4\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{3+1}\\left(θ\\right)$$" ], "result": "=4\\sec^{3+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$3+1=4$$", "result": "=4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7p9w76+/nvHitDrMIupHrrrPmAPxuc/RFHWXkB8A+mnqLeJKmeakO1+Q3aSIldG41PCyA7ixjCdfDYe+yac7KOvOIrkYdKW3Q58EnXaMU+7oslHPSvPf5UfEs5T+IFZ4XHjb2+5NLFZrsH9fcPWg/TXL71sHPAsAQtYZvjpaGSRwboK/Igc1IRC3EGruCoRzV1Zrc/SYRIR8JuxYpD+PJ+Q==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan^{2}\\left(θ\\right)\\right)=2\\tan\\left(θ\\right)\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan^{2}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}2\\tan\\left(θ\\right)\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan^{2}\\left(θ\\right)\\right)", "result": "=2\\tan\\left(θ\\right)\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{2},\\:\\:u=\\tan\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{2}\\right)\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{2}\\right)=2u$$", "input": "\\frac{d}{du}\\left(u^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2u^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2u", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYr+VZUwBnLdzbS6DZQ+f4s+k3hxk9aCfAWodBRxXgUexMchyqTAJWrzJaDbnNcFsJUeCBKuYKgaNJ253gLI69U79qbCA2QqVmvm3jGRXZ2ppvbGT4j1utMEkCDH25m/vlQ==" } }, { "type": "step", "result": "=2u\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\tan\\left(θ\\right)$$", "result": "=2\\tan\\left(θ\\right)\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYux7TmDJ3fopnHGBxoqO+paSXIThcWH58oYoMPt18lzFqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidIn+HfeW3QQwAkRa15J0NSAlj501WLIv46IdbBaParYow2V0WdtbiweShXv1KOwqCjeh7+jKEzLb7VNCEMF3Z/Maswa5oIagKQFOfrnKm5+Gbb/PxX3awut5KxRxyZf8A=" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "result": "=\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYs0Vsx3c0nt53Utu5DgHTe3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK00HV/bLwwglXz/2YNx2i4JHggSrmCoGjSdud4CyOvVOgdQN0k+N3NHNAkEIoLrhB631k80nn+ODwI4ID0kCF5+z7zRz81SqRD02HEUE4Wqh" } }, { "type": "step", "result": "=2\\tan\\left(θ\\right)\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=4\\left(4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{2}\\left(θ\\right)+2\\tan\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Simplify $$4\\left(4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{2}\\left(θ\\right)+2\\tan\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right):{\\quad}4\\left(4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)$$", "input": "4\\left(4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{2}\\left(θ\\right)+2\\tan\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right)", "result": "=4\\left(4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "$$4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{2}\\left(θ\\right)=4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)$$", "input": "4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\tan\\left(θ\\right)\\tan^{2}\\left(θ\\right)=\\:\\tan^{1+2}\\left(θ\\right)$$" ], "result": "=4\\sec^{4}\\left(θ\\right)\\tan^{1+2}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+2=3$$", "result": "=4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7GHerUg4zB8o0EeBO6BKV162cmBc6Js8szDUhC2Hxdqi5HNbiQL8+p2P+VJTsJbU7q47vuWedXv2WUg94ER8IwVADhNIEz39BpXewdhMCBb+bbDkw/o6UIkLjzolgTAsqkuHx4ulZqfxZHIsiQUpV1nyo9elyUFFpcPZW02cAt7EtbJdlgVP3iSyKFg6v1zm6cUwY1HWd9EJWOCb4O+kauMdM6yd/xAZrmYAkuRY8/rYkt3WiGR7ZaCaXvz77bMjS" } }, { "type": "interim", "title": "$$2\\tan\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)=2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "2\\tan\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)=\\:\\sec^{2+4}\\left(θ\\right)$$" ], "result": "=2\\tan\\left(θ\\right)\\sec^{2+4}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+4=6$$", "result": "=2\\tan\\left(θ\\right)\\sec^{6}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7OyotidIxGYYp2AND0rbhNaCGttAzkBWvgG18MKDLv3Rdbcltti6PzkTDSwkmBTY6q47vuWedXv2WUg94ER8IwSIYCdSNn6m2gNv8FkjgS1XBL9WNCz07kx2T4iGijNtQjT/+cTnIKvMqhzKaFBbxvzSei4QZxxrazp/cx7cYZvbVeYdTw2sAeZ9o0nz1uUihamteA8UyY/eLm9jagP57hkwLhsnE4D1C6IGVVz/pqlY=" } }, { "type": "step", "result": "=4\\left(2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)+4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7iza9NvP7nq2MIMeLHhROsvkdSragnZ5K8+WMRM5qjJ4/XhyePE1SthuGDJcaFMnncKXVLLQXMot2gtnZH7qY51d1u0XlM+NdfDNBhSESuvlV00rpv8+ZC6TM10tVCSHsi/RlR31DZv0142/jAcann2ASrhThz5UQhS79xvpIR8tIoCq5NivR6sCY2Sfa33kc8jzDW/n9HorFBpgomiqIe+5byrQDQVCXUD0vH/fvOdxyhd7tjiG+GxQNxDvGkZUlqqyy15Nuz77zqSegKeIT2iKI9Km/ZvzCnLF9eqmklKHqRJDJ/DeTzEayKJpmRlyWQi730E1b9JEo9956Qy3sPLLWww1IW5GUOlAGrcUL2OqwiNrEngO+NNvZ9sqNu+2V" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec^{6}\\left(θ\\right)\\right)=6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{6}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}6\\left(\\sec\\left(θ\\right)\\right)^{5}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{6}\\left(θ\\right)\\right)", "result": "=6\\left(\\sec\\left(θ\\right)\\right)^{5}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{6},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{6}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{6}\\right)=6u^{5}$$", "input": "\\frac{d}{du}\\left(u^{6}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=6u^{6-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=6u^{5}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYgOK76JOSNqGa1B1wU6ozWWk3hxk9aCfAWodBRxXgUexqSoDkuZ4Ab1St3bM8oqKcP8//6/nV5O4fb8Xgwi7mapNFxUvwBeni+JEIFAdbegzHfwjUTnSu4cWqw+8m79+NA==" } }, { "type": "step", "result": "=6u^{5}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=6\\left(\\sec\\left(θ\\right)\\right)^{5}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgBaJTGF5uPwSMzxoSyJNVZsqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKieMSe35zYqdvyntnEYxaQ88jJq7JZFtKgvxzzzuaqTGrebfnOah8gAvLMKOeNMctC5tVYZ/3r82LxPimKUFXjQFStvD8GKDNJJN8qiu0XRcVZAYAvbRKqkRVDaDBHkSBtDwYHkfmHHTze3JXk3LjiJp" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=6\\left(\\sec\\left(θ\\right)\\right)^{5}\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$6\\sec^{5}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "6\\sec^{5}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{5}\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{5+1}\\left(θ\\right)$$" ], "result": "=6\\sec^{5+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$5+1=6$$", "result": "=6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7QMJmgA2Sx2hYpGS5j/h1NLPmAPxuc/RFHWXkB8A+mnqLeJKmeakO1+Q3aSIldG417JiujyaxykLR7Tv0Nr+wOGprXgPFMmP3i5vY2oD+e4YslHPSvPf5UfEs5T+IFZ4XHjb2+5NLFZrsH9fcPWg/TRiIlnZT/wCWlhB0uSVNtvx/uR/EZqc6EbI3tnxSI5001Zrc/SYRIR8JuxYpD+PJ+Q==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=24\\left(4\\left(4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Expand $$4\\left(4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "4\\left(4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=24\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Expand $$4\\left(4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right):{\\quad}16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+8\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "4\\left(4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)", "result": "=16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+8\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)+6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b+c\\right)=ab+ac$$", "secondary": [ "$$a=4,\\:b=4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right),\\:c=2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)$$" ], "result": "=4\\cdot\\:4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+4\\cdot\\:2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "interim", "title": "Simplify $$4\\cdot\\:4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+4\\cdot\\:2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+8\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "4\\cdot\\:4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+4\\cdot\\:2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+8\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:4=16$$", "result": "=16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+4\\cdot\\:2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:2=8$$", "result": "=16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+8\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7iza9NvP7nq2MIMeLHhROsoL8UIMToMV6hDKJhY+Nk7toxrHhSJArgBpsnuAV9UCfBgOJ+lz5D5t+hsCNYCoDMHWD310L1+P2yDQQfMEhENFHx6CvpVa90MvMzklpzJWXZAzoThz1g+TITtf+WJ7KiyRRh7kCo0obc/CTaNyPGkYuuAKnMbBNnPJ2x/QM7ydV1sD7NfhsPe7eDHrmjY0mEyzoLVQPyp9R+5GC0prpwlf2K8DiYaJsjNsjOcdOcdQLW6BryIKCtddVLZsZyZ1dV9DmtpfPTl/o4ZJKfQIdsQqwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "primary": "Add similar elements: $$8\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)+6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)=14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7iza9NvP7nq2MIMeLHhROsoL8UIMToMV6hDKJhY+Nk7toxrHhSJArgBpsnuAV9UCf8SVGtBi1ovwgT5vDB2QfkF3B8WOxYrIgIdHvusvufLAAlilG71elit3w1IBbYN0PumuR+HimzGDTYbsfY1Q6JXyo9elyUFFpcPZW02cAt7EXYiLGBL7PG2xAqJd1w6BXKK1Q1tVX7SUdYWONKsZTX66q9beEAcfrflX+8QRghq2CcvL2jSzw1VlRcGX4ka6qwZ5KNwm43WeyBMBBKL30QvI8w1v5/R6KxQaYKJoqiHsLiDrnJdtlx1I2mhkkUTElN8b/k79QNO9t6sAoPBR3zJYojigZ8i1ymB3KwWBswnjiAEmXhYw7WsDRrfT9tRiW" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=-8\\left(-4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+24\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=-8\\left(-4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+24\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "Evaluate $$-8\\left(-4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+24\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)\\:$$at point $$θ=0:{\\quad}0$$", "result": "=0", "steps": [ { "type": "step", "primary": "Take the point $$θ=0$$ and plug it into $$-8\\left(-4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+24\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)$$", "result": "=-8\\left(-4\\sec^{2}\\left(0\\right)\\tan\\left(0\\right)+12\\sec^{4}\\left(0\\right)\\tan\\left(0\\right)\\right)+24\\left(16\\sec^{4}\\left(0\\right)\\tan^{3}\\left(0\\right)+14\\sec^{6}\\left(0\\right)\\tan\\left(0\\right)\\right)" }, { "type": "interim", "title": "$$8\\left(-4\\sec^{2}\\left(0\\right)\\tan\\left(0\\right)+12\\sec^{4}\\left(0\\right)\\tan\\left(0\\right)\\right)=0$$", "input": "8\\left(-4\\sec^{2}\\left(0\\right)\\tan\\left(0\\right)+12\\sec^{4}\\left(0\\right)\\tan\\left(0\\right)\\right)", "steps": [ { "type": "interim", "title": "$$4\\sec^{2}\\left(0\\right)\\tan\\left(0\\right)=0$$", "input": "4\\sec^{2}\\left(0\\right)\\tan\\left(0\\right)", "steps": [ { "type": "interim", "title": "$$\\sec^{2}\\left(0\\right)=1$$", "input": "\\sec^{2}\\left(0\\right)", "steps": [ { "type": "interim", "title": "Simplify $$\\sec\\left(0\\right):{\\quad}1$$", "input": "\\sec\\left(0\\right)", "result": "=1^{2}", "steps": [ { "type": "step", "primary": "Use the following trivial identity:$${\\quad}\\sec\\left(0\\right)=1$$", "secondary": [ "$$\\sec\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sec(x)&x&\\sec(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{2\\sqrt{3}}{3}&\\frac{7π}{6}&-\\frac{2\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&\\sqrt{2}&\\frac{5π}{4}&-\\sqrt{2}\\\\\\hline \\frac{π}{3}&2&\\frac{4π}{3}&-2\\\\\\hline \\frac{π}{2}&\\mathrm{Undefined}&\\frac{3π}{2}&\\mathrm{Undefined}\\\\\\hline \\frac{2π}{3}&-2&\\frac{5π}{3}&2\\\\\\hline \\frac{3π}{4}&-\\sqrt{2}&\\frac{7π}{4}&\\sqrt{2}\\\\\\hline \\frac{5π}{6}&-\\frac{2\\sqrt{3}}{3}&\\frac{11π}{6}&\\frac{2\\sqrt{3}}{3}\\\\\\hline \\end{array}$$" ], "result": "=1" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7y1xaZeQTSzp609voOcYk6VXTSum/z5kLpMzXS1UJIexiYFOaxxQqg3o8CNKkTm2qW+Sc57RV820557NJqaC71BfVjTo90KMWspnEdE4CKjE=" } }, { "type": "step", "result": "=4\\cdot\\:1\\cdot\\:\\tan\\left(0\\right)" }, { "type": "interim", "title": "Simplify $$\\tan\\left(0\\right):{\\quad}0$$", "input": "\\tan\\left(0\\right)", "result": "=4\\cdot\\:1\\cdot\\:0", "steps": [ { "type": "step", "primary": "Use the following trivial identity:$${\\quad}\\tan\\left(0\\right)=0$$", "secondary": [ "$$\\tan\\left(x\\right)$$ periodicity table with $$πn$$ cycle:<br/>$$\\begin{array}{|c|c|}\\hline x&\\tan(x)\\\\\\hline 0&0\\\\\\hline \\frac{π}{6}&\\frac{\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&1\\\\\\hline \\frac{π}{3}&\\sqrt{3}\\\\\\hline \\frac{π}{2}&\\pm\\infty\\\\\\hline \\frac{2π}{3}&-\\sqrt{3}\\\\\\hline \\frac{3π}{4}&-1\\\\\\hline \\frac{5π}{6}&-\\frac{\\sqrt{3}}{3}\\\\\\hline &\\\\\\hline \\end{array}$$" ], "result": "=0" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Apply rule $$0\\cdot\\:a=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7yoGM/PYNKxD+4Xcq8hjhc6mPj3gXRowY53lqkfZS9MsJQJZuTAY5js+oqjdT8kslKXPrgUnq5rRq9Cvw1ceDci6Tg1fVs3FHX+2oeKw3d9PzoXLK02yt5K+3YXszYC4c" } }, { "type": "interim", "title": "$$12\\sec^{4}\\left(0\\right)\\tan\\left(0\\right)=0$$", "input": "12\\sec^{4}\\left(0\\right)\\tan\\left(0\\right)", "steps": [ { "type": "interim", "title": "$$\\sec^{4}\\left(0\\right)=1$$", "input": "\\sec^{4}\\left(0\\right)", "steps": [ { "type": "interim", "title": "Simplify $$\\sec\\left(0\\right):{\\quad}1$$", "input": "\\sec\\left(0\\right)", "result": "=1^{4}", "steps": [ { "type": "step", "primary": "Use the following trivial identity:$${\\quad}\\sec\\left(0\\right)=1$$", "secondary": [ "$$\\sec\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sec(x)&x&\\sec(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{2\\sqrt{3}}{3}&\\frac{7π}{6}&-\\frac{2\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&\\sqrt{2}&\\frac{5π}{4}&-\\sqrt{2}\\\\\\hline \\frac{π}{3}&2&\\frac{4π}{3}&-2\\\\\\hline \\frac{π}{2}&\\mathrm{Undefined}&\\frac{3π}{2}&\\mathrm{Undefined}\\\\\\hline \\frac{2π}{3}&-2&\\frac{5π}{3}&2\\\\\\hline \\frac{3π}{4}&-\\sqrt{2}&\\frac{7π}{4}&\\sqrt{2}\\\\\\hline \\frac{5π}{6}&-\\frac{2\\sqrt{3}}{3}&\\frac{11π}{6}&\\frac{2\\sqrt{3}}{3}\\\\\\hline \\end{array}$$" ], "result": "=1" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7TnbPBMsEd78kmmwfYDHqNVXTSum/z5kLpMzXS1UJIexiYFOaxxQqg3o8CNKkTm2qW+Sc57RV820557NJqaC71Ka6Ma3rg33eDIhfX/iuVnM=" } }, { "type": "step", "result": "=12\\cdot\\:1\\cdot\\:\\tan\\left(0\\right)" }, { "type": "interim", "title": "Simplify $$\\tan\\left(0\\right):{\\quad}0$$", "input": "\\tan\\left(0\\right)", "result": "=12\\cdot\\:1\\cdot\\:0", "steps": [ { "type": "step", "primary": "Use the following trivial identity:$${\\quad}\\tan\\left(0\\right)=0$$", "secondary": [ "$$\\tan\\left(x\\right)$$ periodicity table with $$πn$$ cycle:<br/>$$\\begin{array}{|c|c|}\\hline x&\\tan(x)\\\\\\hline 0&0\\\\\\hline \\frac{π}{6}&\\frac{\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&1\\\\\\hline \\frac{π}{3}&\\sqrt{3}\\\\\\hline \\frac{π}{2}&\\pm\\infty\\\\\\hline \\frac{2π}{3}&-\\sqrt{3}\\\\\\hline \\frac{3π}{4}&-1\\\\\\hline \\frac{5π}{6}&-\\frac{\\sqrt{3}}{3}\\\\\\hline &\\\\\\hline \\end{array}$$" ], "result": "=0" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Apply rule $$0\\cdot\\:a=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ztKQGqbHeZS/JvOABkqK/RjTuW+Q2J5Q5LUQGIymb9t1g99dC9fj9sg0EHzBIRDRd79UrkSVT0SCLs80Lgihl04wkUSqnBCRad8lKr40KATbzo7SL2lLjXVd+4Xv52R0JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=8\\left(0-0\\right)" }, { "type": "step", "primary": "Add/Subtract the numbers: $$-0+0=0$$", "result": "=8\\cdot\\:0" }, { "type": "step", "primary": "Apply rule $$0\\cdot\\:a=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7iH0ZZWu/57qQnUk5xyzslAnt1nd3Bhlv5dVInbaEGkWwKwfJEYKIuZZNasvHulbS9s43/aG4RqIhR8wn5cy6TATlqoY3+wtl96gVlHC0Yuo/y9DKGIPglJ+qMi9xDu2KX7Nzt2Ly0KJEIhsUV0VhspaQ3x5UBSTkBtTLZJG1WFSzXF1PdZKGLQpjjNSl325EEEWJXNA8Rj59tQEICFmFoA==" } }, { "type": "interim", "title": "$$24\\left(16\\sec^{4}\\left(0\\right)\\tan^{3}\\left(0\\right)+14\\sec^{6}\\left(0\\right)\\tan\\left(0\\right)\\right)=0$$", "input": "24\\left(16\\sec^{4}\\left(0\\right)\\tan^{3}\\left(0\\right)+14\\sec^{6}\\left(0\\right)\\tan\\left(0\\right)\\right)", "steps": [ { "type": "interim", "title": "$$16\\sec^{4}\\left(0\\right)\\tan^{3}\\left(0\\right)=0$$", "input": "16\\sec^{4}\\left(0\\right)\\tan^{3}\\left(0\\right)", "steps": [ { "type": "interim", "title": "$$\\sec^{4}\\left(0\\right)=1$$", "input": "\\sec^{4}\\left(0\\right)", "steps": [ { "type": "interim", "title": "Simplify $$\\sec\\left(0\\right):{\\quad}1$$", "input": "\\sec\\left(0\\right)", "result": "=1^{4}", "steps": [ { "type": "step", "primary": "Use the following trivial identity:$${\\quad}\\sec\\left(0\\right)=1$$", "secondary": [ "$$\\sec\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sec(x)&x&\\sec(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{2\\sqrt{3}}{3}&\\frac{7π}{6}&-\\frac{2\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&\\sqrt{2}&\\frac{5π}{4}&-\\sqrt{2}\\\\\\hline \\frac{π}{3}&2&\\frac{4π}{3}&-2\\\\\\hline \\frac{π}{2}&\\mathrm{Undefined}&\\frac{3π}{2}&\\mathrm{Undefined}\\\\\\hline \\frac{2π}{3}&-2&\\frac{5π}{3}&2\\\\\\hline \\frac{3π}{4}&-\\sqrt{2}&\\frac{7π}{4}&\\sqrt{2}\\\\\\hline \\frac{5π}{6}&-\\frac{2\\sqrt{3}}{3}&\\frac{11π}{6}&\\frac{2\\sqrt{3}}{3}\\\\\\hline \\end{array}$$" ], "result": "=1" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7TnbPBMsEd78kmmwfYDHqNVXTSum/z5kLpMzXS1UJIexiYFOaxxQqg3o8CNKkTm2qW+Sc57RV820557NJqaC71Ka6Ma3rg33eDIhfX/iuVnM=" } }, { "type": "step", "result": "=16\\cdot\\:1\\cdot\\:\\tan^{3}\\left(0\\right)" }, { "type": "interim", "title": "$$\\tan^{3}\\left(0\\right)=0$$", "input": "\\tan^{3}\\left(0\\right)", "steps": [ { "type": "interim", "title": "Simplify $$\\tan\\left(0\\right):{\\quad}0$$", "input": "\\tan\\left(0\\right)", "result": "=0^{3}", "steps": [ { "type": "step", "primary": "Use the following trivial identity:$${\\quad}\\tan\\left(0\\right)=0$$", "secondary": [ "$$\\tan\\left(x\\right)$$ periodicity table with $$πn$$ cycle:<br/>$$\\begin{array}{|c|c|}\\hline x&\\tan(x)\\\\\\hline 0&0\\\\\\hline \\frac{π}{6}&\\frac{\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&1\\\\\\hline \\frac{π}{3}&\\sqrt{3}\\\\\\hline \\frac{π}{2}&\\pm\\infty\\\\\\hline \\frac{2π}{3}&-\\sqrt{3}\\\\\\hline \\frac{3π}{4}&-1\\\\\\hline \\frac{5π}{6}&-\\frac{\\sqrt{3}}{3}\\\\\\hline &\\\\\\hline \\end{array}$$" ], "result": "=0" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Apply rule $$0^{a}=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7GE9JoQhnOmDIpJSiT8FGMFXTSum/z5kLpMzXS1UJIeyqjfZqhkrOoAz8sRb7wXZWPGHkI0MFUiTEwg/g9Re5WAQEEAYC9h/Nb4xd/kIOxRQ=" } }, { "type": "step", "result": "=16\\cdot\\:1\\cdot\\:0" }, { "type": "step", "primary": "Apply rule $$0\\cdot\\:a=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7AhXsSCpYfSFLoEvOM6cQkr/Ehopp9rGqiIQLEoitJ5IgJ/ZZA32ZInFBpDtxBfiKf7LqB9CcyvYCWDsGseX09k29cjXxPd/F95KFhHS4hjheH6PVg9CLFp1ktSOp93NiCFX7kPfEkIuYfqIFW07ybg==" } }, { "type": "interim", "title": "$$14\\sec^{6}\\left(0\\right)\\tan\\left(0\\right)=0$$", "input": "14\\sec^{6}\\left(0\\right)\\tan\\left(0\\right)", "steps": [ { "type": "interim", "title": "$$\\sec^{6}\\left(0\\right)=1$$", "input": "\\sec^{6}\\left(0\\right)", "steps": [ { "type": "interim", "title": "Simplify $$\\sec\\left(0\\right):{\\quad}1$$", "input": "\\sec\\left(0\\right)", "result": "=1^{6}", "steps": [ { "type": "step", "primary": "Use the following trivial identity:$${\\quad}\\sec\\left(0\\right)=1$$", "secondary": [ "$$\\sec\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sec(x)&x&\\sec(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{2\\sqrt{3}}{3}&\\frac{7π}{6}&-\\frac{2\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&\\sqrt{2}&\\frac{5π}{4}&-\\sqrt{2}\\\\\\hline \\frac{π}{3}&2&\\frac{4π}{3}&-2\\\\\\hline \\frac{π}{2}&\\mathrm{Undefined}&\\frac{3π}{2}&\\mathrm{Undefined}\\\\\\hline \\frac{2π}{3}&-2&\\frac{5π}{3}&2\\\\\\hline \\frac{3π}{4}&-\\sqrt{2}&\\frac{7π}{4}&\\sqrt{2}\\\\\\hline \\frac{5π}{6}&-\\frac{2\\sqrt{3}}{3}&\\frac{11π}{6}&\\frac{2\\sqrt{3}}{3}\\\\\\hline \\end{array}$$" ], "result": "=1" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7WcSYqbVPn/qsbXXyqHdMbFXTSum/z5kLpMzXS1UJIexiYFOaxxQqg3o8CNKkTm2qW+Sc57RV820557NJqaC71Idwe88IvH72vjqcga7gyBg=" } }, { "type": "step", "result": "=14\\cdot\\:1\\cdot\\:\\tan\\left(0\\right)" }, { "type": "interim", "title": "Simplify $$\\tan\\left(0\\right):{\\quad}0$$", "input": "\\tan\\left(0\\right)", "result": "=14\\cdot\\:1\\cdot\\:0", "steps": [ { "type": "step", "primary": "Use the following trivial identity:$${\\quad}\\tan\\left(0\\right)=0$$", "secondary": [ "$$\\tan\\left(x\\right)$$ periodicity table with $$πn$$ cycle:<br/>$$\\begin{array}{|c|c|}\\hline x&\\tan(x)\\\\\\hline 0&0\\\\\\hline \\frac{π}{6}&\\frac{\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&1\\\\\\hline \\frac{π}{3}&\\sqrt{3}\\\\\\hline \\frac{π}{2}&\\pm\\infty\\\\\\hline \\frac{2π}{3}&-\\sqrt{3}\\\\\\hline \\frac{3π}{4}&-1\\\\\\hline \\frac{5π}{6}&-\\frac{\\sqrt{3}}{3}\\\\\\hline &\\\\\\hline \\end{array}$$" ], "result": "=0" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Apply rule $$0\\cdot\\:a=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7uESehsDSxkrMlWGVYh0nEhjTuW+Q2J5Q5LUQGIymb9t1g99dC9fj9sg0EHzBIRDRd79UrkSVT0SCLs80Lgihl/MHZz/q3kUqOBwvqGYpXX/bzo7SL2lLjXVd+4Xv52R0JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=24\\left(0+0\\right)" }, { "type": "step", "primary": "Add the numbers: $$0+0=0$$", "result": "=24\\cdot\\:0" }, { "type": "step", "primary": "Apply rule $$0\\cdot\\:a=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s76nlao4YNedUk2S4zzCJvQLvrZMykvfhAOYSGWz3LDed1Xm1xDLBN/I9hH5mwedz1q3Z1IfyMnAKE4AfR1MbC4glAlm5MBjmOz6iqN1PySyUpc+uBSermtGr0K/DVx4NyXbuwsGATDsU7VXcOrctTY9xzzDZrUHthRg6PFIysvL6g1+/Xb7zhBvdvyquoZxFY+mnfm1+QDeBe9hfT/VE9PA==" } }, { "type": "step", "result": "=-0+0" }, { "type": "step", "primary": "Simplify" }, { "type": "step", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Laplace Eval Function At Point Specific 3Eq" } }, { "type": "step", "result": "0" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\frac{d^{7}}{dθ^{7}}\\left(\\tan\\left(θ\\right)\\right)\\left(0\\right){\\quad:\\quad}272$$", "input": "\\frac{d^{7}}{dθ^{7}}\\left(\\tan\\left(θ\\right)\\right)\\left(0\\right)", "steps": [ { "type": "interim", "title": "$$\\frac{d^{7}}{dθ^{7}}\\left(\\tan\\left(θ\\right)\\right)=-8\\left(-4\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)+12\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)\\right)+24\\left(64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+132\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+14\\sec^{8}\\left(θ\\right)\\right)$$", "input": "\\frac{d^{7}}{dθ^{7}}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "result": "=\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYs0Vsx3c0nt53Utu5DgHTe3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK00HV/bLwwglXz/2YNx2i4JHggSrmCoGjSdud4CyOvVOgdQN0k+N3NHNAkEIoLrhB631k80nn+ODwI4ID0kCF5+z7zRz81SqRD02HEUE4Wqh" } }, { "type": "step", "result": "=\\frac{d^{6}}{dθ^{6}}\\left(\\sec^{2}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{2},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{2}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{2}\\right)=2u$$", "input": "\\frac{d}{du}\\left(u^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2u^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2u", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYr+VZUwBnLdzbS6DZQ+f4s+k3hxk9aCfAWodBRxXgUexMchyqTAJWrzJaDbnNcFsJUeCBKuYKgaNJ253gLI69U79qbCA2QqVmvm3jGRXZ2ppvbGT4j1utMEkCDH25m/vlQ==" } }, { "type": "step", "result": "=2u\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgCSXIThcWH58oYoMPt18lzFqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidZClirlWNFPnHR1rvdyoFLI7IC+oqnwb1T6nPLpiw1eysw2NXPd8jZnr3fcPpqc7ejeh7+jKEzLb7VNCEMF3Z/Maswa5oIagKQFOfrnKm5+Gbb/PxX3awut5KxRxyZf8A=" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{1+1}\\left(θ\\right)$$" ], "result": "=2\\sec^{1+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7aY4rYekMZhUfP6sZ6mfqd6Lsz0lt1XLEr88Kc/cjJo5V00rpv8+ZC6TM10tVCSHspqrCcTwb/I0f8OwMjuU6eBAUa+AHsHd5OGpMiuFITjOuqvW3hAHH635V/vEEYIatHimBRYRqHSWeJkuUPhfTC5LQ0B1JNgsb5SXzbS+SCnAfGB9H25Zc+Ua1z7Kw8kgb4gBJl4WMO1rA0a30/bUYlg==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{d^{5}}{dθ^{5}}\\left(2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\right)=-4\\sec^{2}\\left(θ\\right)+6\\sec^{4}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=2\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the Product Rule: $$\\left(f{\\cdot}g\\right)'=f'{\\cdot}g+f{\\cdot}g'$$", "secondary": [ "$$f=\\sec^{2}\\left(θ\\right),\\:g=\\tan\\left(θ\\right)$$" ], "result": "=2\\left(\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)\\tan\\left(θ\\right)+\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)\\sec^{2}\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Product%20Rule", "practiceTopic": "Product Rule" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{2},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{2}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{2}\\right)=2u$$", "input": "\\frac{d}{du}\\left(u^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2u^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2u", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYr+VZUwBnLdzbS6DZQ+f4s+k3hxk9aCfAWodBRxXgUexMchyqTAJWrzJaDbnNcFsJUeCBKuYKgaNJ253gLI69U79qbCA2QqVmvm3jGRXZ2ppvbGT4j1utMEkCDH25m/vlQ==" } }, { "type": "step", "result": "=2u\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgCSXIThcWH58oYoMPt18lzFqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidZClirlWNFPnHR1rvdyoFLI7IC+oqnwb1T6nPLpiw1eysw2NXPd8jZnr3fcPpqc7ejeh7+jKEzLb7VNCEMF3Z/Maswa5oIagKQFOfrnKm5+Gbb/PxX3awut5KxRxyZf8A=" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{1+1}\\left(θ\\right)$$" ], "result": "=2\\sec^{1+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7aY4rYekMZhUfP6sZ6mfqd6Lsz0lt1XLEr88Kc/cjJo5V00rpv8+ZC6TM10tVCSHspqrCcTwb/I0f8OwMjuU6eBAUa+AHsHd5OGpMiuFITjOuqvW3hAHH635V/vEEYIatHimBRYRqHSWeJkuUPhfTC5LQ0B1JNgsb5SXzbS+SCnAfGB9H25Zc+Ua1z7Kw8kgb4gBJl4WMO1rA0a30/bUYlg==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "result": "=\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYs0Vsx3c0nt53Utu5DgHTe3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK00HV/bLwwglXz/2YNx2i4JHggSrmCoGjSdud4CyOvVOgdQN0k+N3NHNAkEIoLrhB631k80nn+ODwI4ID0kCF5+z7zRz81SqRD02HEUE4Wqh" } }, { "type": "step", "result": "=2\\left(2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Simplify $$2\\left(2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\right):{\\quad}-4\\sec^{2}\\left(θ\\right)+6\\sec^{4}\\left(θ\\right)$$", "input": "2\\left(2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\right)", "result": "=-4\\sec^{2}\\left(θ\\right)+6\\sec^{4}\\left(θ\\right)", "steps": [ { "type": "interim", "title": "$$2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)=2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)$$", "input": "2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\tan\\left(θ\\right)\\tan\\left(θ\\right)=\\:\\tan^{1+1}\\left(θ\\right)$$" ], "result": "=2\\sec^{2}\\left(θ\\right)\\tan^{1+1}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Md1SJKHYLxhOYRRbsLTu+axUZgt6FTmFSHnRsotkqemLeJKmeakO1+Q3aSIldG41s7BwSVdKEYSqn+4fKAQoGk0xmFIl1KTO5aAkhaHvKzBALhShbMGZIZsZWTO721HsWI4dG63F/aSz/PFDvu5FnO3sngYNzcydIedGQjjtM8sBMUhkLMqDD6KHvgZAgKMdRQzIIgXxnWeDhDTk2Lfum8RPUSGriaFRn0NZXHZrPQw=" } }, { "type": "interim", "title": "$$\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)=\\sec^{4}\\left(θ\\right)$$", "input": "\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)=\\:\\sec^{2+2}\\left(θ\\right)$$" ], "result": "=\\sec^{2+2}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+2=4$$", "result": "=\\sec^{4}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7AtrsReczPdUqIdn5/Ev5mFtNfVJqszzpMwg6spHO8F8gJ/ZZA32ZInFBpDtxBfiKn3Jyj2VJgBojKQHjINpkkETYvnjqwhFu6aWmDOYpAVMO/U6qyRZonn1WPQ/21pz3cKXVLLQXMot2gtnZH7qY5zkbLioo7YVO0SRnDAO/7Uokt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "=2\\left(\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Expand $$\\left(\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)\\cdot\\:2:{\\quad}2\\sec^{4}\\left(θ\\right)+4\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)$$", "input": "\\left(\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)\\cdot\\:2", "result": "=2\\sec^{4}\\left(θ\\right)+4\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)", "steps": [ { "type": "step", "result": "=2\\left(\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the distributive law: $$a\\left(b+c\\right)=ab+ac$$", "secondary": [ "$$a=2,\\:b=\\sec^{4}\\left(θ\\right),\\:c=2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)$$" ], "result": "=2\\sec^{4}\\left(θ\\right)+2\\cdot\\:2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=2\\sec^{4}\\left(θ\\right)+4\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7cTiUO1fOFax2wXHZewjhxEUAdh/O2PLAxzMjSpKce9YL4tWjzbYCZhBh/Hc2rUXacMKakGXZAry+N/hvJOodjXCQoYlYQ8U+Tfyx0kyzI8iF3F2TkMs7mKjIsUPAR/Vd82dH/qvnCC7/oZy4HZlX2aHh6acbPp8G70jj/skoFx7vbBmbuQNTF0TphKZ8Ruvacu+LCkBvFQsw7xBL2Ys2TUYb1899n2cPVSJTSzzdZzp+38s+uQz1W6FLDk3KkIYWhJuRFkG6Ai089lDjeB1lMbCI2sSeA74029n2yo277ZU=" } }, { "type": "interim", "title": "Rewrite using trig identities", "input": "2\\sec^{4}\\left(θ\\right)+4\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)", "result": "=-4\\sec^{2}\\left(θ\\right)+6\\sec^{4}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Use the Pythagorean identity: $$\\tan^{2}\\left(x\\right)+1=\\sec^{2}\\left(x\\right)$$", "secondary": [ "$$\\tan^{2}\\left(x\\right)=\\sec^{2}\\left(x\\right)-1$$" ], "result": "=2\\sec^{4}\\left(θ\\right)+4\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right)" }, { "type": "interim", "title": "Simplify $$2\\sec^{4}\\left(θ\\right)+4\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right):{\\quad}6\\sec^{4}\\left(θ\\right)-4\\sec^{2}\\left(θ\\right)$$", "input": "2\\sec^{4}\\left(θ\\right)+4\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right)", "result": "=6\\sec^{4}\\left(θ\\right)-4\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "interim", "title": "Expand $$4\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right):{\\quad}4\\sec^{4}\\left(θ\\right)-4\\sec^{2}\\left(θ\\right)$$", "input": "4\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right)", "result": "=2\\sec^{4}\\left(θ\\right)+4\\sec^{4}\\left(θ\\right)-4\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b-c\\right)=ab-ac$$", "secondary": [ "$$a=4\\sec^{2}\\left(θ\\right),\\:b=\\sec^{2}\\left(θ\\right),\\:c=1$$" ], "result": "=4\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)-4\\sec^{2}\\left(θ\\right)\\cdot\\:1", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "result": "=4\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)-4\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$4\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)-4\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right):{\\quad}4\\sec^{4}\\left(θ\\right)-4\\sec^{2}\\left(θ\\right)$$", "input": "4\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)-4\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right)", "result": "=4\\sec^{4}\\left(θ\\right)-4\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "interim", "title": "$$4\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)=4\\sec^{4}\\left(θ\\right)$$", "input": "4\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)=\\:\\sec^{2+2}\\left(θ\\right)$$" ], "result": "=4\\sec^{2+2}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+2=4$$", "result": "=4\\sec^{4}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vjDr4p6TvcgiSLV1uDo4U9JYPDPDesonRI520G8zYRPehkKrn0era9rz8TlL+x/vMHVZf8xiryCPkbUIYXhiIHRfFgXB+vlRb/AHCJklpF9e7YF3TFAJSjcLX2VvhbubqX/cSuhK5Ty2xqd/IdNr6zlZ8CVoKZl2BAoDdAtcFNxU2fg2Eo8khoQfnK/P3ock" } }, { "type": "interim", "title": "$$4\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right)=4\\sec^{2}\\left(θ\\right)$$", "input": "4\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:1=4$$", "result": "=4\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7xF9rlQkuAOrRzKTf6eQs7fgkIqVBCzsTrV5i/ECNCw8tOtZYwUjyXhDTsNnn6Elr/BCsllMMEyc8UGKoUwuYxu6vfsn/ogOG+fHMfNQh9JeZfoRPsbCjih+xtMTOMGl4tzgLeG7CLgj8aHpc44iME/LWpEthEIQrAjNHY4qDKt/ET1Ehq4mhUZ9DWVx2az0M" } }, { "type": "step", "result": "=4\\sec^{4}\\left(θ\\right)-4\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vjDr4p6TvcgiSLV1uDo4U/jbMbEoNNbIR/7Q84OynCfTLx8mOdHYVzxX643JqKFIvTZkYD4gUeSoFrhGGhxq98kwYX4VRPb1geKkn+NU7mcE3SiHEhDlxIXJDVY+PPdvgQUxJPyUNnGfVirkcwpVO84fwxlvni/T8DIWnD0mw9U6FFV/0Hm6Y1hI7VaPzzKhkSoZP7CyArJlOo0FRny79Q==" } }, { "type": "step", "primary": "Add similar elements: $$2\\sec^{4}\\left(θ\\right)+4\\sec^{4}\\left(θ\\right)=6\\sec^{4}\\left(θ\\right)$$", "result": "=6\\sec^{4}\\left(θ\\right)-4\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7f48emNsIl4MI4hQZuZje4penFTHzh6X+DX3uvPdqYy7umM/GfBKdAtyuCOAeIKZmAJYpRu9XpYrd8NSAW2DdD0c/CXknyRKFbCevkP3vkuVyZXAw9UlCE0mEZtdsFPrV7q9+yf+iA4b58cx81CH0l/C30sSftAIFS6Qkpy19Ikp0x8irWzKvvppKMUvqTrnrUObZozYPTEmgGDR5V3grHDoUVX/QebpjWEjtVo/PMqGRKhk/sLICsmU6jQVGfLv1" } } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7xqDNX42OyJjPevbF+7IBng7708BkbVE1gb8/0kMEgzvXUIQv0aG2ncbDgSWX/DWycNZx/2EDub2dSsLw3NGg9xLC2T8NFZvTA+pff5AdXPdX9lRN4mlYq/zI0dgxEXGdPvdnhi0s3lZ+DJ7rEiTeqenxKTmAB1ahPFE542Dd/TeutOLi0kB5xHysaBsHIgntIaTy8gmKy5PY3el+RbUmlkmFP65WPUHCIGTpzT/EAccetg60+ZUlqP5jB+ReldUA5AMRxXWt+t9va0qZyagxyQ==" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7xI7dknI212svHgZHHIcoRvkdSragnZ5K8+WMRM5qjJ6ETkUamdUz59t0aHpV3uN0rE0Ib0rKQ2oM97vl/gDbowCWKUbvV6WK3fDUgFtg3Q+MfZ1BFRhhGnQf9q28g9nK/SzNF20Xrftd883pI0RALQ5+VkRmueXOP55Piqx5MyWBBTEk/JQ2cZ9WKuRzClU7m4RppR4RRNJR6+bqv/lHExUnakC9Y1jhr4uI97E9yvalDk6K3L00/zgNSSliALTtnpy3tAot7+5PIP9fBKZw2Llxun299aFzbjCyRFblUKE=" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{d^{4}}{dθ^{4}}\\left(-4\\sec^{2}\\left(θ\\right)+6\\sec^{4}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(-4\\sec^{2}\\left(θ\\right)+6\\sec^{4}\\left(θ\\right)\\right)=-8\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+24\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(-4\\sec^{2}\\left(θ\\right)+6\\sec^{4}\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=-\\frac{d}{dθ}\\left(4\\sec^{2}\\left(θ\\right)\\right)+\\frac{d}{dθ}\\left(6\\sec^{4}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(4\\sec^{2}\\left(θ\\right)\\right)=8\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(4\\sec^{2}\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=4\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Apply the chain rule:$${\\quad}2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{2},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{2}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{2}\\right)=2u$$", "input": "\\frac{d}{du}\\left(u^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2u^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2u", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYr+VZUwBnLdzbS6DZQ+f4s+k3hxk9aCfAWodBRxXgUexMchyqTAJWrzJaDbnNcFsJUeCBKuYKgaNJ253gLI69U79qbCA2QqVmvm3jGRXZ2ppvbGT4j1utMEkCDH25m/vlQ==" } }, { "type": "step", "result": "=2u\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgCSXIThcWH58oYoMPt18lzFqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidZClirlWNFPnHR1rvdyoFLI7IC+oqnwb1T6nPLpiw1eysw2NXPd8jZnr3fcPpqc7ejeh7+jKEzLb7VNCEMF3Z/Maswa5oIagKQFOfrnKm5+Gbb/PxX3awut5KxRxyZf8A=" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=4\\cdot\\:2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$4\\cdot\\:2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}8\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "4\\cdot\\:2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=8\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:2=8$$", "result": "=8\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{1+1}\\left(θ\\right)$$" ], "result": "=8\\sec^{1+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=8\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7NqBYpvGKZC5lMRQos2qph5K/zREn/U8meKQ3TvRjTQz8cZMsH3IX6lxp+ANHGbuCq47vuWedXv2WUg94ER8IwTfgzf+5OuhkULh+nunjrqae0Rv083m3K3ie3DVArOu08LfSxJ+0AgVLpCSnLX0iSqXiJoCK22T5YMsRbcSz8DsyFZaieHAvmtaw5r6RKguDHxgfR9uWXPlGtc+ysPJIG+IASZeFjDtawNGt9P21GJY=" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(6\\sec^{4}\\left(θ\\right)\\right)=24\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(6\\sec^{4}\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=6\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Apply the chain rule:$${\\quad}4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{4},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{4}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{4}\\right)=4u^{3}$$", "input": "\\frac{d}{du}\\left(u^{4}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=4u^{4-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=4u^{3}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYl3oEe4EC6n4B8Xgz0cKnkqk3hxk9aCfAWodBRxXgUexuzo0Fl6VhneutEteidn+Kv8//6/nV5O4fb8Xgwi7mapNFxUvwBeni+JEIFAdbegzYEOcQgyW4n51cR7AeMAm7Q==" } }, { "type": "step", "result": "=4u^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgA2CO5cm49xNEqFV0DMaZJRqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidVduWEnoDYDg06IbaTsp9TFtqiuNhFTvmF8ZCDqWZNoubfnOah8gAvLMKOeNMctC5tVYZ/3r82LxPimKUFXjQFStvD8GKDNJJN8qiu0XRcVZAYAvbRKqkRVDaDBHkSBtDwYHkfmHHTze3JXk3LjiJp" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=6\\cdot\\:4\\left(\\sec\\left(θ\\right)\\right)^{3}\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$6\\cdot\\:4\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}24\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "6\\cdot\\:4\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=24\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$6\\cdot\\:4=24$$", "result": "=24\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{3+1}\\left(θ\\right)$$" ], "result": "=24\\sec^{3+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$3+1=4$$", "result": "=24\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l3FkS70/4BFsebxcaZbHnTNSoP5XQt+n/5y371e8QHJROlTYN926lBwmiJdD5HUSzRqDxPUzBN6vjj5oJL9kUIdu2MPseykkfgn/Trt10KGp9ej8WEAztP3UNEC9n7jjZEt3ZXAiqUE0HIXrrrezJHZKl2mV4AlztiApTAFigz07AW6Jz0sXkPNp6b6xpHMqM1Kg/ldC36f/nLfvV7xActytFlzCV8oq55bHLwXbb0c=" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=-8\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+24\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{d^{3}}{dθ^{3}}\\left(-8\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+24\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(-8\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+24\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)=-8\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)+24\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(-8\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+24\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=-\\frac{d}{dθ}\\left(8\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+\\frac{d}{dθ}\\left(24\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(8\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\right)=8\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(8\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=8\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the Product Rule: $$\\left(f{\\cdot}g\\right)'=f'{\\cdot}g+f{\\cdot}g'$$", "secondary": [ "$$f=\\sec^{2}\\left(θ\\right),\\:g=\\tan\\left(θ\\right)$$" ], "result": "=8\\left(\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)\\tan\\left(θ\\right)+\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)\\sec^{2}\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Product%20Rule", "practiceTopic": "Product Rule" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{2},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{2}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{2}\\right)=2u$$", "input": "\\frac{d}{du}\\left(u^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2u^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2u", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYr+VZUwBnLdzbS6DZQ+f4s+k3hxk9aCfAWodBRxXgUexMchyqTAJWrzJaDbnNcFsJUeCBKuYKgaNJ253gLI69U79qbCA2QqVmvm3jGRXZ2ppvbGT4j1utMEkCDH25m/vlQ==" } }, { "type": "step", "result": "=2u\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgCSXIThcWH58oYoMPt18lzFqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidZClirlWNFPnHR1rvdyoFLI7IC+oqnwb1T6nPLpiw1eysw2NXPd8jZnr3fcPpqc7ejeh7+jKEzLb7VNCEMF3Z/Maswa5oIagKQFOfrnKm5+Gbb/PxX3awut5KxRxyZf8A=" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{1+1}\\left(θ\\right)$$" ], "result": "=2\\sec^{1+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7aY4rYekMZhUfP6sZ6mfqd6Lsz0lt1XLEr88Kc/cjJo5V00rpv8+ZC6TM10tVCSHspqrCcTwb/I0f8OwMjuU6eBAUa+AHsHd5OGpMiuFITjOuqvW3hAHH635V/vEEYIatHimBRYRqHSWeJkuUPhfTC5LQ0B1JNgsb5SXzbS+SCnAfGB9H25Zc+Ua1z7Kw8kgb4gBJl4WMO1rA0a30/bUYlg==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "result": "=\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYs0Vsx3c0nt53Utu5DgHTe3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK00HV/bLwwglXz/2YNx2i4JHggSrmCoGjSdud4CyOvVOgdQN0k+N3NHNAkEIoLrhB631k80nn+ODwI4ID0kCF5+z7zRz81SqRD02HEUE4Wqh" } }, { "type": "step", "result": "=8\\left(2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Simplify $$8\\left(2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\right):{\\quad}8\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)$$", "input": "8\\left(2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\right)", "result": "=8\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "$$2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)=2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)$$", "input": "2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\tan\\left(θ\\right)\\tan\\left(θ\\right)=\\:\\tan^{1+1}\\left(θ\\right)$$" ], "result": "=2\\sec^{2}\\left(θ\\right)\\tan^{1+1}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Md1SJKHYLxhOYRRbsLTu+axUZgt6FTmFSHnRsotkqemLeJKmeakO1+Q3aSIldG41s7BwSVdKEYSqn+4fKAQoGk0xmFIl1KTO5aAkhaHvKzBALhShbMGZIZsZWTO721HsWI4dG63F/aSz/PFDvu5FnO3sngYNzcydIedGQjjtM8sBMUhkLMqDD6KHvgZAgKMdRQzIIgXxnWeDhDTk2Lfum8RPUSGriaFRn0NZXHZrPQw=" } }, { "type": "interim", "title": "$$\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)=\\sec^{4}\\left(θ\\right)$$", "input": "\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)=\\:\\sec^{2+2}\\left(θ\\right)$$" ], "result": "=\\sec^{2+2}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+2=4$$", "result": "=\\sec^{4}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7AtrsReczPdUqIdn5/Ev5mFtNfVJqszzpMwg6spHO8F8gJ/ZZA32ZInFBpDtxBfiKn3Jyj2VJgBojKQHjINpkkETYvnjqwhFu6aWmDOYpAVMO/U6qyRZonn1WPQ/21pz3cKXVLLQXMot2gtnZH7qY5zkbLioo7YVO0SRnDAO/7Uokt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "=8\\left(\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Simplify $$2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{4}\\left(θ\\right):{\\quad}3\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)$$", "input": "2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{4}\\left(θ\\right)", "result": "=8\\left(3\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Rewrite using trig identities", "input": "\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)", "result": "=-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Use the Pythagorean identity: $$\\tan^{2}\\left(x\\right)+1=\\sec^{2}\\left(x\\right)$$", "secondary": [ "$$\\tan^{2}\\left(x\\right)=\\sec^{2}\\left(x\\right)-1$$" ], "result": "=\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right)" }, { "type": "interim", "title": "Simplify $$\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right):{\\quad}3\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)$$", "input": "\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right)", "result": "=3\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "interim", "title": "Expand $$2\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right):{\\quad}2\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)$$", "input": "2\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right)", "result": "=\\sec^{4}\\left(θ\\right)+2\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b-c\\right)=ab-ac$$", "secondary": [ "$$a=2\\sec^{2}\\left(θ\\right),\\:b=\\sec^{2}\\left(θ\\right),\\:c=1$$" ], "result": "=2\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)\\cdot\\:1", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "result": "=2\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)-2\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$2\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)-2\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right):{\\quad}2\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)$$", "input": "2\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)-2\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right)", "result": "=2\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "interim", "title": "$$2\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)=2\\sec^{4}\\left(θ\\right)$$", "input": "2\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)=\\:\\sec^{2+2}\\left(θ\\right)$$" ], "result": "=2\\sec^{2+2}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+2=4$$", "result": "=2\\sec^{4}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Md1SJKHYLxhOYRRbsLTu+dJYPDPDesonRI520G8zYRPehkKrn0era9rz8TlL+x/vyvn14PieOyuPP98CkiPIaXRfFgXB+vlRb/AHCJklpF+qQC22PXYzQL8KzAhJCzszqX/cSuhK5Ty2xqd/IdNr67A2LXxYB8f3C3KpWaLgll1U2fg2Eo8khoQfnK/P3ock" } }, { "type": "interim", "title": "$$2\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right)=2\\sec^{2}\\left(θ\\right)$$", "input": "2\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=2\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ht1jgkupea8+jUgT+M/VFPgkIqVBCzsTrV5i/ECNCw8tOtZYwUjyXhDTsNnn6Elr8U8lPA4O6WXezAN1J2P7Ue6vfsn/ogOG+fHMfNQh9Je7opJQAzFYPpqn/2bLavXutzgLeG7CLgj8aHpc44iMEx91+u64Nzpp2/441mWGcQ/ET1Ehq4mhUZ9DWVx2az0M" } }, { "type": "step", "result": "=2\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Md1SJKHYLxhOYRRbsLTu+fjbMbEoNNbIR/7Q84OynCfTLx8mOdHYVzxX643JqKFI1Jetp9AvoQcY2QSDwFJhz1kUvGSawlk7umQ8kurx5doE3SiHEhDlxIXJDVY+PPdvgQUxJPyUNnGfVirkcwpVO6RMkE3rByM/Y1Dxxfs3xFw6FFV/0Hm6Y1hI7VaPzzKhkSoZP7CyArJlOo0FRny79Q==" } }, { "type": "step", "primary": "Add similar elements: $$\\sec^{4}\\left(θ\\right)+2\\sec^{4}\\left(θ\\right)=3\\sec^{4}\\left(θ\\right)$$", "result": "=3\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ARkhlGZdazrdxNLzhlqePtRiXXTzenxbt00ahL9sDS5L0MjjUe01mdeSwRMoYPSBLTrWWMFI8l4Q07DZ5+hJazaxvejhvTJvqoS105+T9+xc3qH/hVuCay9igw/1IwZBNySLUGdPDJeW3uY1XSs479bA+zX4bD3u3gx65o2NJhP8KqIsxz0lwScBMkZS31XhRCRwG0yFaMyhS3uC5/uUeMvqafR00cq/gMXQBnztqPn0yTzBsNnlyCSr/zrcxdQG" } } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7FkSOkw7Nc+ISjlBZ7XiESO/dTLaj0YrNSw6+/ogyGX8k+jBZ4XxujMMeS3Qvsi6m7aEe0hvT9VtxXei653t2zQi9Vnn4imguIoWn3N1eqjdGIkb3QW2w7y4N83RSktbc/kUupyp7xXxB8YmKEQ5HoPs4SAUa1zTfp2gkDWnxoYVtNtNgAtA4xHxju0w39zFkZEt3ZXAiqUE0HIXrrrezJJi0cdkgnw2HPjRVpcHMxZbIciYVzjaAolEaNkVONtBBRppTNgMFons9bMkTzIHooA==" } } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7JzuoJyZ0ket+0by442o6nfkdSragnZ5K8+WMRM5qjJ6ETkUamdUz59t0aHpV3uN0rE0Ib0rKQ2oM97vl/gDbowCWKUbvV6WK3fDUgFtg3Q+nr1JqgQ+t+tehVxpQusgo+5k7zGGz9wmA2lyQsVWy+j6F2h9EjPuS9/S594y1Nd1FKk3fejFkyiOiq9iG9IkAIuSmJUDtLxRPyJ57Nkq7qjUyCLMvQZ782OaKpAVh7FOsVGYLehU5hUh50bKLZKnpo1d4a+pBCGznOTLxP2sni3enTnyZLeuZs6pBwuQMplUkt3WiGR7ZaCaXvz77bMjS" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(24\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)=24\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(24\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=24\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the Product Rule: $$\\left(f{\\cdot}g\\right)'=f'{\\cdot}g+f{\\cdot}g'$$", "secondary": [ "$$f=\\sec^{4}\\left(θ\\right),\\:g=\\tan\\left(θ\\right)$$" ], "result": "=24\\left(\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)\\tan\\left(θ\\right)+\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)\\sec^{4}\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Product%20Rule", "practiceTopic": "Product Rule" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)=4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{4},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{4}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{4}\\right)=4u^{3}$$", "input": "\\frac{d}{du}\\left(u^{4}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=4u^{4-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=4u^{3}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYl3oEe4EC6n4B8Xgz0cKnkqk3hxk9aCfAWodBRxXgUexuzo0Fl6VhneutEteidn+Kv8//6/nV5O4fb8Xgwi7mapNFxUvwBeni+JEIFAdbegzYEOcQgyW4n51cR7AeMAm7Q==" } }, { "type": "step", "result": "=4u^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgA2CO5cm49xNEqFV0DMaZJRqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidVduWEnoDYDg06IbaTsp9TFtqiuNhFTvmF8ZCDqWZNoubfnOah8gAvLMKOeNMctC5tVYZ/3r82LxPimKUFXjQFStvD8GKDNJJN8qiu0XRcVZAYAvbRKqkRVDaDBHkSBtDwYHkfmHHTze3JXk3LjiJp" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$4\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "4\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{3+1}\\left(θ\\right)$$" ], "result": "=4\\sec^{3+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$3+1=4$$", "result": "=4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7p9w76+/nvHitDrMIupHrrrPmAPxuc/RFHWXkB8A+mnqLeJKmeakO1+Q3aSIldG41PCyA7ixjCdfDYe+yac7KOvOIrkYdKW3Q58EnXaMU+7oslHPSvPf5UfEs5T+IFZ4XHjb2+5NLFZrsH9fcPWg/TXL71sHPAsAQtYZvjpaGSRwboK/Igc1IRC3EGruCoRzV1Zrc/SYRIR8JuxYpD+PJ+Q==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "result": "=\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYs0Vsx3c0nt53Utu5DgHTe3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK00HV/bLwwglXz/2YNx2i4JHggSrmCoGjSdud4CyOvVOgdQN0k+N3NHNAkEIoLrhB631k80nn+ODwI4ID0kCF5+z7zRz81SqRD02HEUE4Wqh" } }, { "type": "step", "result": "=24\\left(4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Simplify $$24\\left(4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right):{\\quad}24\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)$$", "input": "24\\left(4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right)", "result": "=24\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "$$4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)=4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)$$", "input": "4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\tan\\left(θ\\right)\\tan\\left(θ\\right)=\\:\\tan^{1+1}\\left(θ\\right)$$" ], "result": "=4\\sec^{4}\\left(θ\\right)\\tan^{1+1}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7GHerUg4zB8o0EeBO6BKV16xUZgt6FTmFSHnRsotkqemLeJKmeakO1+Q3aSIldG41PCyA7ixjCdfDYe+yac7KOvOIrkYdKW3Q58EnXaMU+7pALhShbMGZIZsZWTO721HsXaBT4KNKuhZZSFsFWnKRStdNJOkRdCtrWB4iMQVmMwBoAWE3NiWlLPrDNnXLOZLK3tKI9zGyi4xCTaccmboyHcRPUSGriaFRn0NZXHZrPQw=" } }, { "type": "interim", "title": "$$\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)=\\sec^{6}\\left(θ\\right)$$", "input": "\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)=\\:\\sec^{2+4}\\left(θ\\right)$$" ], "result": "=\\sec^{2+4}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+4=6$$", "result": "=\\sec^{6}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7AtrsReczPdUqIdn5/Ev5mLBxohAhN47qWD2nlwOHqfkgJ/ZZA32ZInFBpDtxBfiKn3Jyj2VJgBojKQHjINpkkG6JHhxn2McSW7OxLgQBi64O/U6qyRZonn1WPQ/21pz3xMe3sR8ZMoA2Ynlp2W853t3OcQiSW6rZ4zI81UhADI4kt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "=24\\left(\\sec^{6}\\left(θ\\right)+4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7jhDHfABKmSqcJJ1fAFAsGBiMJmjxigW8LortZKvrqEbWPF6oWuXggupsFvXrooIqxMe3sR8ZMoA2Ynlp2W853tMvHyY50dhXPFfrjcmooUg3CXpyb+Iu3cMrCn545PU8uO1NenZJP60hEwBH05kmlvOIrkYdKW3Q58EnXaMU+7r4ZAjrJM4mwvgfYovsxoopgQUxJPyUNnGfVirkcwpVOxKnUVO0TgjLT50Mmk0DzLYh0PvFonSnuZi7Z6ItIhmYhYbzXwn7iok10o2w+NJLRiZIb9CFuBJdPyi+MBoLCMd/LnYygZMYhjzpUCXiHwCy" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=-8\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)+24\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{d^{2}}{dθ^{2}}\\left(-8\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)+24\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(-8\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)+24\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)\\right)=-8\\left(-4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+24\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(-8\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)+24\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=-\\frac{d}{dθ}\\left(8\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)\\right)+\\frac{d}{dθ}\\left(24\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(8\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)\\right)=8\\left(-4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(8\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=8\\frac{d}{dθ}\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=8\\left(-\\frac{d}{dθ}\\left(2\\sec^{2}\\left(θ\\right)\\right)+\\frac{d}{dθ}\\left(3\\sec^{4}\\left(θ\\right)\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(2\\sec^{2}\\left(θ\\right)\\right)=4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(2\\sec^{2}\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=2\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Apply the chain rule:$${\\quad}2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{2},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{2}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{2}\\right)=2u$$", "input": "\\frac{d}{du}\\left(u^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2u^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2u", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYr+VZUwBnLdzbS6DZQ+f4s+k3hxk9aCfAWodBRxXgUexMchyqTAJWrzJaDbnNcFsJUeCBKuYKgaNJ253gLI69U79qbCA2QqVmvm3jGRXZ2ppvbGT4j1utMEkCDH25m/vlQ==" } }, { "type": "step", "result": "=2u\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgCSXIThcWH58oYoMPt18lzFqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidZClirlWNFPnHR1rvdyoFLI7IC+oqnwb1T6nPLpiw1eysw2NXPd8jZnr3fcPpqc7ejeh7+jKEzLb7VNCEMF3Z/Maswa5oIagKQFOfrnKm5+Gbb/PxX3awut5KxRxyZf8A=" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=2\\cdot\\:2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$2\\cdot\\:2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "2\\cdot\\:2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=4\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{1+1}\\left(θ\\right)$$" ], "result": "=4\\sec^{1+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s71jEHNygbGF2qQyJrTxQFvpK/zREn/U8meKQ3TvRjTQz8cZMsH3IX6lxp+ANHGbuCq47vuWedXv2WUg94ER8IwfulLd9uW7xFGDkqRxB3vSSe0Rv083m3K3ie3DVArOu08LfSxJ+0AgVLpCSnLX0iSnTHyKtbMq++mkoxS+pOuesyFZaieHAvmtaw5r6RKguDHxgfR9uWXPlGtc+ysPJIG+IASZeFjDtawNGt9P21GJY=" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(3\\sec^{4}\\left(θ\\right)\\right)=12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(3\\sec^{4}\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=3\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Apply the chain rule:$${\\quad}4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{4},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{4}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{4}\\right)=4u^{3}$$", "input": "\\frac{d}{du}\\left(u^{4}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=4u^{4-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=4u^{3}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYl3oEe4EC6n4B8Xgz0cKnkqk3hxk9aCfAWodBRxXgUexuzo0Fl6VhneutEteidn+Kv8//6/nV5O4fb8Xgwi7mapNFxUvwBeni+JEIFAdbegzYEOcQgyW4n51cR7AeMAm7Q==" } }, { "type": "step", "result": "=4u^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgA2CO5cm49xNEqFV0DMaZJRqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidVduWEnoDYDg06IbaTsp9TFtqiuNhFTvmF8ZCDqWZNoubfnOah8gAvLMKOeNMctC5tVYZ/3r82LxPimKUFXjQFStvD8GKDNJJN8qiu0XRcVZAYAvbRKqkRVDaDBHkSBtDwYHkfmHHTze3JXk3LjiJp" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=3\\cdot\\:4\\left(\\sec\\left(θ\\right)\\right)^{3}\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$3\\cdot\\:4\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "3\\cdot\\:4\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$3\\cdot\\:4=12$$", "result": "=12\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{3+1}\\left(θ\\right)$$" ], "result": "=12\\sec^{3+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$3+1=4$$", "result": "=12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7jFGb0q9vd+kdWvVYeHJHbDNSoP5XQt+n/5y371e8QHJROlTYN926lBwmiJdD5HUSzRqDxPUzBN6vjj5oJL9kUFPzsE52Mw7P00gxEE61byKp9ej8WEAztP3UNEC9n7jjZEt3ZXAiqUE0HIXrrrezJHZKl2mV4AlztiApTAFigz2gNkWvAT/c3O3uDO/qxcKwM1Kg/ldC36f/nLfvV7xActytFlzCV8oq55bHLwXbb0c=" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=8\\left(-4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(24\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)\\right)=24\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(24\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=24\\frac{d}{dθ}\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=24\\left(\\frac{d}{dθ}\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)+\\frac{d}{dθ}\\left(\\sec^{6}\\left(θ\\right)\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)=4\\left(4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=4\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the Product Rule: $$\\left(f{\\cdot}g\\right)'=f'{\\cdot}g+f{\\cdot}g'$$", "secondary": [ "$$f=\\sec^{4}\\left(θ\\right),\\:g=\\tan^{2}\\left(θ\\right)$$" ], "result": "=4\\left(\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)\\tan^{2}\\left(θ\\right)+\\frac{d}{dθ}\\left(\\tan^{2}\\left(θ\\right)\\right)\\sec^{4}\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Product%20Rule", "practiceTopic": "Product Rule" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)=4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{4},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{4}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{4}\\right)=4u^{3}$$", "input": "\\frac{d}{du}\\left(u^{4}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=4u^{4-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=4u^{3}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYl3oEe4EC6n4B8Xgz0cKnkqk3hxk9aCfAWodBRxXgUexuzo0Fl6VhneutEteidn+Kv8//6/nV5O4fb8Xgwi7mapNFxUvwBeni+JEIFAdbegzYEOcQgyW4n51cR7AeMAm7Q==" } }, { "type": "step", "result": "=4u^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgA2CO5cm49xNEqFV0DMaZJRqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidVduWEnoDYDg06IbaTsp9TFtqiuNhFTvmF8ZCDqWZNoubfnOah8gAvLMKOeNMctC5tVYZ/3r82LxPimKUFXjQFStvD8GKDNJJN8qiu0XRcVZAYAvbRKqkRVDaDBHkSBtDwYHkfmHHTze3JXk3LjiJp" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$4\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "4\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{3+1}\\left(θ\\right)$$" ], "result": "=4\\sec^{3+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$3+1=4$$", "result": "=4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7p9w76+/nvHitDrMIupHrrrPmAPxuc/RFHWXkB8A+mnqLeJKmeakO1+Q3aSIldG41PCyA7ixjCdfDYe+yac7KOvOIrkYdKW3Q58EnXaMU+7oslHPSvPf5UfEs5T+IFZ4XHjb2+5NLFZrsH9fcPWg/TXL71sHPAsAQtYZvjpaGSRwboK/Igc1IRC3EGruCoRzV1Zrc/SYRIR8JuxYpD+PJ+Q==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan^{2}\\left(θ\\right)\\right)=2\\tan\\left(θ\\right)\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan^{2}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}2\\tan\\left(θ\\right)\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan^{2}\\left(θ\\right)\\right)", "result": "=2\\tan\\left(θ\\right)\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{2},\\:\\:u=\\tan\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{2}\\right)\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{2}\\right)=2u$$", "input": "\\frac{d}{du}\\left(u^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2u^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2u", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYr+VZUwBnLdzbS6DZQ+f4s+k3hxk9aCfAWodBRxXgUexMchyqTAJWrzJaDbnNcFsJUeCBKuYKgaNJ253gLI69U79qbCA2QqVmvm3jGRXZ2ppvbGT4j1utMEkCDH25m/vlQ==" } }, { "type": "step", "result": "=2u\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\tan\\left(θ\\right)$$", "result": "=2\\tan\\left(θ\\right)\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYux7TmDJ3fopnHGBxoqO+paSXIThcWH58oYoMPt18lzFqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidIn+HfeW3QQwAkRa15J0NSAlj501WLIv46IdbBaParYow2V0WdtbiweShXv1KOwqCjeh7+jKEzLb7VNCEMF3Z/Maswa5oIagKQFOfrnKm5+Gbb/PxX3awut5KxRxyZf8A=" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "result": "=\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYs0Vsx3c0nt53Utu5DgHTe3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK00HV/bLwwglXz/2YNx2i4JHggSrmCoGjSdud4CyOvVOgdQN0k+N3NHNAkEIoLrhB631k80nn+ODwI4ID0kCF5+z7zRz81SqRD02HEUE4Wqh" } }, { "type": "step", "result": "=2\\tan\\left(θ\\right)\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=4\\left(4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{2}\\left(θ\\right)+2\\tan\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Simplify $$4\\left(4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{2}\\left(θ\\right)+2\\tan\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right):{\\quad}4\\left(4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)$$", "input": "4\\left(4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{2}\\left(θ\\right)+2\\tan\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right)", "result": "=4\\left(4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "$$4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{2}\\left(θ\\right)=4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)$$", "input": "4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\tan\\left(θ\\right)\\tan^{2}\\left(θ\\right)=\\:\\tan^{1+2}\\left(θ\\right)$$" ], "result": "=4\\sec^{4}\\left(θ\\right)\\tan^{1+2}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+2=3$$", "result": "=4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7GHerUg4zB8o0EeBO6BKV162cmBc6Js8szDUhC2Hxdqi5HNbiQL8+p2P+VJTsJbU7q47vuWedXv2WUg94ER8IwVADhNIEz39BpXewdhMCBb+bbDkw/o6UIkLjzolgTAsqkuHx4ulZqfxZHIsiQUpV1nyo9elyUFFpcPZW02cAt7EtbJdlgVP3iSyKFg6v1zm6cUwY1HWd9EJWOCb4O+kauMdM6yd/xAZrmYAkuRY8/rYkt3WiGR7ZaCaXvz77bMjS" } }, { "type": "interim", "title": "$$2\\tan\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)=2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "2\\tan\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)=\\:\\sec^{2+4}\\left(θ\\right)$$" ], "result": "=2\\tan\\left(θ\\right)\\sec^{2+4}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+4=6$$", "result": "=2\\tan\\left(θ\\right)\\sec^{6}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7OyotidIxGYYp2AND0rbhNaCGttAzkBWvgG18MKDLv3Rdbcltti6PzkTDSwkmBTY6q47vuWedXv2WUg94ER8IwSIYCdSNn6m2gNv8FkjgS1XBL9WNCz07kx2T4iGijNtQjT/+cTnIKvMqhzKaFBbxvzSei4QZxxrazp/cx7cYZvbVeYdTw2sAeZ9o0nz1uUihamteA8UyY/eLm9jagP57hkwLhsnE4D1C6IGVVz/pqlY=" } }, { "type": "step", "result": "=4\\left(2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)+4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7iza9NvP7nq2MIMeLHhROsvkdSragnZ5K8+WMRM5qjJ4/XhyePE1SthuGDJcaFMnncKXVLLQXMot2gtnZH7qY51d1u0XlM+NdfDNBhSESuvlV00rpv8+ZC6TM10tVCSHsi/RlR31DZv0142/jAcann2ASrhThz5UQhS79xvpIR8tIoCq5NivR6sCY2Sfa33kc8jzDW/n9HorFBpgomiqIe+5byrQDQVCXUD0vH/fvOdxyhd7tjiG+GxQNxDvGkZUlqqyy15Nuz77zqSegKeIT2iKI9Km/ZvzCnLF9eqmklKHqRJDJ/DeTzEayKJpmRlyWQi730E1b9JEo9956Qy3sPLLWww1IW5GUOlAGrcUL2OqwiNrEngO+NNvZ9sqNu+2V" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec^{6}\\left(θ\\right)\\right)=6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{6}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}6\\left(\\sec\\left(θ\\right)\\right)^{5}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{6}\\left(θ\\right)\\right)", "result": "=6\\left(\\sec\\left(θ\\right)\\right)^{5}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{6},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{6}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{6}\\right)=6u^{5}$$", "input": "\\frac{d}{du}\\left(u^{6}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=6u^{6-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=6u^{5}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYgOK76JOSNqGa1B1wU6ozWWk3hxk9aCfAWodBRxXgUexqSoDkuZ4Ab1St3bM8oqKcP8//6/nV5O4fb8Xgwi7mapNFxUvwBeni+JEIFAdbegzHfwjUTnSu4cWqw+8m79+NA==" } }, { "type": "step", "result": "=6u^{5}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=6\\left(\\sec\\left(θ\\right)\\right)^{5}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgBaJTGF5uPwSMzxoSyJNVZsqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKieMSe35zYqdvyntnEYxaQ88jJq7JZFtKgvxzzzuaqTGrebfnOah8gAvLMKOeNMctC5tVYZ/3r82LxPimKUFXjQFStvD8GKDNJJN8qiu0XRcVZAYAvbRKqkRVDaDBHkSBtDwYHkfmHHTze3JXk3LjiJp" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=6\\left(\\sec\\left(θ\\right)\\right)^{5}\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$6\\sec^{5}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "6\\sec^{5}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{5}\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{5+1}\\left(θ\\right)$$" ], "result": "=6\\sec^{5+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$5+1=6$$", "result": "=6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7QMJmgA2Sx2hYpGS5j/h1NLPmAPxuc/RFHWXkB8A+mnqLeJKmeakO1+Q3aSIldG417JiujyaxykLR7Tv0Nr+wOGprXgPFMmP3i5vY2oD+e4YslHPSvPf5UfEs5T+IFZ4XHjb2+5NLFZrsH9fcPWg/TRiIlnZT/wCWlhB0uSVNtvx/uR/EZqc6EbI3tnxSI5001Zrc/SYRIR8JuxYpD+PJ+Q==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=24\\left(4\\left(4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Expand $$4\\left(4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "4\\left(4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=24\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Expand $$4\\left(4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right):{\\quad}16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+8\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "4\\left(4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)", "result": "=16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+8\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)+6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b+c\\right)=ab+ac$$", "secondary": [ "$$a=4,\\:b=4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right),\\:c=2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)$$" ], "result": "=4\\cdot\\:4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+4\\cdot\\:2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "interim", "title": "Simplify $$4\\cdot\\:4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+4\\cdot\\:2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+8\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "4\\cdot\\:4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+4\\cdot\\:2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+8\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:4=16$$", "result": "=16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+4\\cdot\\:2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:2=8$$", "result": "=16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+8\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7iza9NvP7nq2MIMeLHhROsoL8UIMToMV6hDKJhY+Nk7toxrHhSJArgBpsnuAV9UCfBgOJ+lz5D5t+hsCNYCoDMHWD310L1+P2yDQQfMEhENFHx6CvpVa90MvMzklpzJWXZAzoThz1g+TITtf+WJ7KiyRRh7kCo0obc/CTaNyPGkYuuAKnMbBNnPJ2x/QM7ydV1sD7NfhsPe7eDHrmjY0mEyzoLVQPyp9R+5GC0prpwlf2K8DiYaJsjNsjOcdOcdQLW6BryIKCtddVLZsZyZ1dV9DmtpfPTl/o4ZJKfQIdsQqwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "primary": "Add similar elements: $$8\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)+6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)=14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7iza9NvP7nq2MIMeLHhROsoL8UIMToMV6hDKJhY+Nk7toxrHhSJArgBpsnuAV9UCf8SVGtBi1ovwgT5vDB2QfkF3B8WOxYrIgIdHvusvufLAAlilG71elit3w1IBbYN0PumuR+HimzGDTYbsfY1Q6JXyo9elyUFFpcPZW02cAt7EXYiLGBL7PG2xAqJd1w6BXKK1Q1tVX7SUdYWONKsZTX66q9beEAcfrflX+8QRghq2CcvL2jSzw1VlRcGX4ka6qwZ5KNwm43WeyBMBBKL30QvI8w1v5/R6KxQaYKJoqiHsLiDrnJdtlx1I2mhkkUTElN8b/k79QNO9t6sAoPBR3zJYojigZ8i1ymB3KwWBswnjiAEmXhYw7WsDRrfT9tRiW" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=-8\\left(-4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+24\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{d}{dθ}\\left(-8\\left(-4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+24\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(-8\\left(-4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+24\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)\\right)=-8\\left(-4\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)+12\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)\\right)+24\\left(64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+132\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+14\\sec^{8}\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(-8\\left(-4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+24\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=-\\frac{d}{dθ}\\left(8\\left(-4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)\\right)+\\frac{d}{dθ}\\left(24\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(8\\left(-4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)\\right)=8\\left(-4\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)+12\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(8\\left(-4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=8\\frac{d}{dθ}\\left(-4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=8\\left(-\\frac{d}{dθ}\\left(4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+\\frac{d}{dθ}\\left(12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\right)=4\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=4\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the Product Rule: $$\\left(f{\\cdot}g\\right)'=f'{\\cdot}g+f{\\cdot}g'$$", "secondary": [ "$$f=\\sec^{2}\\left(θ\\right),\\:g=\\tan\\left(θ\\right)$$" ], "result": "=4\\left(\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)\\tan\\left(θ\\right)+\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)\\sec^{2}\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Product%20Rule", "practiceTopic": "Product Rule" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{2},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{2}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{2}\\right)=2u$$", "input": "\\frac{d}{du}\\left(u^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2u^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2u", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYr+VZUwBnLdzbS6DZQ+f4s+k3hxk9aCfAWodBRxXgUexMchyqTAJWrzJaDbnNcFsJUeCBKuYKgaNJ253gLI69U79qbCA2QqVmvm3jGRXZ2ppvbGT4j1utMEkCDH25m/vlQ==" } }, { "type": "step", "result": "=2u\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgCSXIThcWH58oYoMPt18lzFqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidZClirlWNFPnHR1rvdyoFLI7IC+oqnwb1T6nPLpiw1eysw2NXPd8jZnr3fcPpqc7ejeh7+jKEzLb7VNCEMF3Z/Maswa5oIagKQFOfrnKm5+Gbb/PxX3awut5KxRxyZf8A=" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{1+1}\\left(θ\\right)$$" ], "result": "=2\\sec^{1+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7aY4rYekMZhUfP6sZ6mfqd6Lsz0lt1XLEr88Kc/cjJo5V00rpv8+ZC6TM10tVCSHspqrCcTwb/I0f8OwMjuU6eBAUa+AHsHd5OGpMiuFITjOuqvW3hAHH635V/vEEYIatHimBRYRqHSWeJkuUPhfTC5LQ0B1JNgsb5SXzbS+SCnAfGB9H25Zc+Ua1z7Kw8kgb4gBJl4WMO1rA0a30/bUYlg==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "result": "=\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYs0Vsx3c0nt53Utu5DgHTe3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK00HV/bLwwglXz/2YNx2i4JHggSrmCoGjSdud4CyOvVOgdQN0k+N3NHNAkEIoLrhB631k80nn+ODwI4ID0kCF5+z7zRz81SqRD02HEUE4Wqh" } }, { "type": "step", "result": "=4\\left(2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Simplify $$4\\left(2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\right):{\\quad}4\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)$$", "input": "4\\left(2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\right)", "result": "=4\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "$$2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)=2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)$$", "input": "2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\tan\\left(θ\\right)\\tan\\left(θ\\right)=\\:\\tan^{1+1}\\left(θ\\right)$$" ], "result": "=2\\sec^{2}\\left(θ\\right)\\tan^{1+1}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Md1SJKHYLxhOYRRbsLTu+axUZgt6FTmFSHnRsotkqemLeJKmeakO1+Q3aSIldG41s7BwSVdKEYSqn+4fKAQoGk0xmFIl1KTO5aAkhaHvKzBALhShbMGZIZsZWTO721HsWI4dG63F/aSz/PFDvu5FnO3sngYNzcydIedGQjjtM8sBMUhkLMqDD6KHvgZAgKMdRQzIIgXxnWeDhDTk2Lfum8RPUSGriaFRn0NZXHZrPQw=" } }, { "type": "interim", "title": "$$\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)=\\sec^{4}\\left(θ\\right)$$", "input": "\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)=\\:\\sec^{2+2}\\left(θ\\right)$$" ], "result": "=\\sec^{2+2}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+2=4$$", "result": "=\\sec^{4}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7AtrsReczPdUqIdn5/Ev5mFtNfVJqszzpMwg6spHO8F8gJ/ZZA32ZInFBpDtxBfiKn3Jyj2VJgBojKQHjINpkkETYvnjqwhFu6aWmDOYpAVMO/U6qyRZonn1WPQ/21pz3cKXVLLQXMot2gtnZH7qY5zkbLioo7YVO0SRnDAO/7Uokt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "=4\\left(\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Simplify $$2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{4}\\left(θ\\right):{\\quad}3\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)$$", "input": "2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{4}\\left(θ\\right)", "result": "=4\\left(3\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Rewrite using trig identities", "input": "\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)", "result": "=-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Use the Pythagorean identity: $$\\tan^{2}\\left(x\\right)+1=\\sec^{2}\\left(x\\right)$$", "secondary": [ "$$\\tan^{2}\\left(x\\right)=\\sec^{2}\\left(x\\right)-1$$" ], "result": "=\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right)" }, { "type": "interim", "title": "Simplify $$\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right):{\\quad}3\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)$$", "input": "\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right)", "result": "=3\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "interim", "title": "Expand $$2\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right):{\\quad}2\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)$$", "input": "2\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right)", "result": "=\\sec^{4}\\left(θ\\right)+2\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b-c\\right)=ab-ac$$", "secondary": [ "$$a=2\\sec^{2}\\left(θ\\right),\\:b=\\sec^{2}\\left(θ\\right),\\:c=1$$" ], "result": "=2\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)\\cdot\\:1", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "result": "=2\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)-2\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$2\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)-2\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right):{\\quad}2\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)$$", "input": "2\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)-2\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right)", "result": "=2\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "interim", "title": "$$2\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)=2\\sec^{4}\\left(θ\\right)$$", "input": "2\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)=\\:\\sec^{2+2}\\left(θ\\right)$$" ], "result": "=2\\sec^{2+2}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+2=4$$", "result": "=2\\sec^{4}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Md1SJKHYLxhOYRRbsLTu+dJYPDPDesonRI520G8zYRPehkKrn0era9rz8TlL+x/vyvn14PieOyuPP98CkiPIaXRfFgXB+vlRb/AHCJklpF+qQC22PXYzQL8KzAhJCzszqX/cSuhK5Ty2xqd/IdNr67A2LXxYB8f3C3KpWaLgll1U2fg2Eo8khoQfnK/P3ock" } }, { "type": "interim", "title": "$$2\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right)=2\\sec^{2}\\left(θ\\right)$$", "input": "2\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=2\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ht1jgkupea8+jUgT+M/VFPgkIqVBCzsTrV5i/ECNCw8tOtZYwUjyXhDTsNnn6Elr8U8lPA4O6WXezAN1J2P7Ue6vfsn/ogOG+fHMfNQh9Je7opJQAzFYPpqn/2bLavXutzgLeG7CLgj8aHpc44iMEx91+u64Nzpp2/441mWGcQ/ET1Ehq4mhUZ9DWVx2az0M" } }, { "type": "step", "result": "=2\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Md1SJKHYLxhOYRRbsLTu+fjbMbEoNNbIR/7Q84OynCfTLx8mOdHYVzxX643JqKFI1Jetp9AvoQcY2QSDwFJhz1kUvGSawlk7umQ8kurx5doE3SiHEhDlxIXJDVY+PPdvgQUxJPyUNnGfVirkcwpVO6RMkE3rByM/Y1Dxxfs3xFw6FFV/0Hm6Y1hI7VaPzzKhkSoZP7CyArJlOo0FRny79Q==" } }, { "type": "step", "primary": "Add similar elements: $$\\sec^{4}\\left(θ\\right)+2\\sec^{4}\\left(θ\\right)=3\\sec^{4}\\left(θ\\right)$$", "result": "=3\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ARkhlGZdazrdxNLzhlqePtRiXXTzenxbt00ahL9sDS5L0MjjUe01mdeSwRMoYPSBLTrWWMFI8l4Q07DZ5+hJazaxvejhvTJvqoS105+T9+xc3qH/hVuCay9igw/1IwZBNySLUGdPDJeW3uY1XSs479bA+zX4bD3u3gx65o2NJhP8KqIsxz0lwScBMkZS31XhRCRwG0yFaMyhS3uC5/uUeMvqafR00cq/gMXQBnztqPn0yTzBsNnlyCSr/zrcxdQG" } } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7FkSOkw7Nc+ISjlBZ7XiESO/dTLaj0YrNSw6+/ogyGX8k+jBZ4XxujMMeS3Qvsi6m7aEe0hvT9VtxXei653t2zQi9Vnn4imguIoWn3N1eqjdGIkb3QW2w7y4N83RSktbc/kUupyp7xXxB8YmKEQ5HoPs4SAUa1zTfp2gkDWnxoYVtNtNgAtA4xHxju0w39zFkZEt3ZXAiqUE0HIXrrrezJJi0cdkgnw2HPjRVpcHMxZbIciYVzjaAolEaNkVONtBBRppTNgMFons9bMkTzIHooA==" } } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7aTuTexD/CTl68iqGJang3fkdSragnZ5K8+WMRM5qjJ6ETkUamdUz59t0aHpV3uN0rE0Ib0rKQ2oM97vl/gDbowCWKUbvV6WK3fDUgFtg3Q8XyqIsLZjT7Df8hUObtCZb+5k7zGGz9wmA2lyQsVWy+j6F2h9EjPuS9/S594y1Nd1FKk3fejFkyiOiq9iG9IkAIuSmJUDtLxRPyJ57Nkq7qlZOK6n0rXJ0Nc64aGIzHiSsVGYLehU5hUh50bKLZKnpo1d4a+pBCGznOTLxP2sni3enTnyZLeuZs6pBwuQMplUkt3WiGR7ZaCaXvz77bMjS" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)=12\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=12\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the Product Rule: $$\\left(f{\\cdot}g\\right)'=f'{\\cdot}g+f{\\cdot}g'$$", "secondary": [ "$$f=\\sec^{4}\\left(θ\\right),\\:g=\\tan\\left(θ\\right)$$" ], "result": "=12\\left(\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)\\tan\\left(θ\\right)+\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)\\sec^{4}\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Product%20Rule", "practiceTopic": "Product Rule" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)=4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{4},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{4}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{4}\\right)=4u^{3}$$", "input": "\\frac{d}{du}\\left(u^{4}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=4u^{4-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=4u^{3}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYl3oEe4EC6n4B8Xgz0cKnkqk3hxk9aCfAWodBRxXgUexuzo0Fl6VhneutEteidn+Kv8//6/nV5O4fb8Xgwi7mapNFxUvwBeni+JEIFAdbegzYEOcQgyW4n51cR7AeMAm7Q==" } }, { "type": "step", "result": "=4u^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgA2CO5cm49xNEqFV0DMaZJRqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidVduWEnoDYDg06IbaTsp9TFtqiuNhFTvmF8ZCDqWZNoubfnOah8gAvLMKOeNMctC5tVYZ/3r82LxPimKUFXjQFStvD8GKDNJJN8qiu0XRcVZAYAvbRKqkRVDaDBHkSBtDwYHkfmHHTze3JXk3LjiJp" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$4\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "4\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{3+1}\\left(θ\\right)$$" ], "result": "=4\\sec^{3+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$3+1=4$$", "result": "=4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7p9w76+/nvHitDrMIupHrrrPmAPxuc/RFHWXkB8A+mnqLeJKmeakO1+Q3aSIldG41PCyA7ixjCdfDYe+yac7KOvOIrkYdKW3Q58EnXaMU+7oslHPSvPf5UfEs5T+IFZ4XHjb2+5NLFZrsH9fcPWg/TXL71sHPAsAQtYZvjpaGSRwboK/Igc1IRC3EGruCoRzV1Zrc/SYRIR8JuxYpD+PJ+Q==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "result": "=\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYs0Vsx3c0nt53Utu5DgHTe3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK00HV/bLwwglXz/2YNx2i4JHggSrmCoGjSdud4CyOvVOgdQN0k+N3NHNAkEIoLrhB631k80nn+ODwI4ID0kCF5+z7zRz81SqRD02HEUE4Wqh" } }, { "type": "step", "result": "=12\\left(4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Simplify $$12\\left(4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right):{\\quad}12\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)$$", "input": "12\\left(4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right)", "result": "=12\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "$$4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)=4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)$$", "input": "4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\tan\\left(θ\\right)\\tan\\left(θ\\right)=\\:\\tan^{1+1}\\left(θ\\right)$$" ], "result": "=4\\sec^{4}\\left(θ\\right)\\tan^{1+1}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7GHerUg4zB8o0EeBO6BKV16xUZgt6FTmFSHnRsotkqemLeJKmeakO1+Q3aSIldG41PCyA7ixjCdfDYe+yac7KOvOIrkYdKW3Q58EnXaMU+7pALhShbMGZIZsZWTO721HsXaBT4KNKuhZZSFsFWnKRStdNJOkRdCtrWB4iMQVmMwBoAWE3NiWlLPrDNnXLOZLK3tKI9zGyi4xCTaccmboyHcRPUSGriaFRn0NZXHZrPQw=" } }, { "type": "interim", "title": "$$\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)=\\sec^{6}\\left(θ\\right)$$", "input": "\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)=\\:\\sec^{2+4}\\left(θ\\right)$$" ], "result": "=\\sec^{2+4}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+4=6$$", "result": "=\\sec^{6}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7AtrsReczPdUqIdn5/Ev5mLBxohAhN47qWD2nlwOHqfkgJ/ZZA32ZInFBpDtxBfiKn3Jyj2VJgBojKQHjINpkkG6JHhxn2McSW7OxLgQBi64O/U6qyRZonn1WPQ/21pz3xMe3sR8ZMoA2Ynlp2W853t3OcQiSW6rZ4zI81UhADI4kt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "=12\\left(\\sec^{6}\\left(θ\\right)+4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7lv52tOJZUIL6vqIybdOpThiMJmjxigW8LortZKvrqEbWPF6oWuXggupsFvXrooIqxMe3sR8ZMoA2Ynlp2W853tMvHyY50dhXPFfrjcmooUinnaTA6rKFuYRlEeyjdJSquO1NenZJP60hEwBH05kmlvOIrkYdKW3Q58EnXaMU+7r4ZAjrJM4mwvgfYovsxoopgQUxJPyUNnGfVirkcwpVO48SyOdzKmRYNaCIczIhnoIh0PvFonSnuZi7Z6ItIhmYhYbzXwn7iok10o2w+NJLRiZIb9CFuBJdPyi+MBoLCMd/LnYygZMYhjzpUCXiHwCy" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=8\\left(-4\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)+12\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(24\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)\\right)=24\\left(64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+132\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+14\\sec^{8}\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(24\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=24\\frac{d}{dθ}\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=24\\left(\\frac{d}{dθ}\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)\\right)+\\frac{d}{dθ}\\left(14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)\\right)=16\\left(4\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+3\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=16\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the Product Rule: $$\\left(f{\\cdot}g\\right)'=f'{\\cdot}g+f{\\cdot}g'$$", "secondary": [ "$$f=\\sec^{4}\\left(θ\\right),\\:g=\\tan^{3}\\left(θ\\right)$$" ], "result": "=16\\left(\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)\\tan^{3}\\left(θ\\right)+\\frac{d}{dθ}\\left(\\tan^{3}\\left(θ\\right)\\right)\\sec^{4}\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Product%20Rule", "practiceTopic": "Product Rule" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)=4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{4},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{4}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{4}\\right)=4u^{3}$$", "input": "\\frac{d}{du}\\left(u^{4}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=4u^{4-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=4u^{3}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYl3oEe4EC6n4B8Xgz0cKnkqk3hxk9aCfAWodBRxXgUexuzo0Fl6VhneutEteidn+Kv8//6/nV5O4fb8Xgwi7mapNFxUvwBeni+JEIFAdbegzYEOcQgyW4n51cR7AeMAm7Q==" } }, { "type": "step", "result": "=4u^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgA2CO5cm49xNEqFV0DMaZJRqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidVduWEnoDYDg06IbaTsp9TFtqiuNhFTvmF8ZCDqWZNoubfnOah8gAvLMKOeNMctC5tVYZ/3r82LxPimKUFXjQFStvD8GKDNJJN8qiu0XRcVZAYAvbRKqkRVDaDBHkSBtDwYHkfmHHTze3JXk3LjiJp" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$4\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "4\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{3+1}\\left(θ\\right)$$" ], "result": "=4\\sec^{3+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$3+1=4$$", "result": "=4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7p9w76+/nvHitDrMIupHrrrPmAPxuc/RFHWXkB8A+mnqLeJKmeakO1+Q3aSIldG41PCyA7ixjCdfDYe+yac7KOvOIrkYdKW3Q58EnXaMU+7oslHPSvPf5UfEs5T+IFZ4XHjb2+5NLFZrsH9fcPWg/TXL71sHPAsAQtYZvjpaGSRwboK/Igc1IRC3EGruCoRzV1Zrc/SYRIR8JuxYpD+PJ+Q==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan^{3}\\left(θ\\right)\\right)=3\\tan^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan^{3}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}3\\left(\\tan\\left(θ\\right)\\right)^{2}\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan^{3}\\left(θ\\right)\\right)", "result": "=3\\left(\\tan\\left(θ\\right)\\right)^{2}\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{3},\\:\\:u=\\tan\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{3}\\right)\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{3}\\right)=3u^{2}$$", "input": "\\frac{d}{du}\\left(u^{3}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=3u^{3-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=3u^{2}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYjNrXZy15fg+DDm4IZ/kmJKk3hxk9aCfAWodBRxXgUexYrhQEWwJJMG//0JI2CMZRv8//6/nV5O4fb8Xgwi7mapNFxUvwBeni+JEIFAdbegz8nAGF3XMCO2i5cp0uYL6iA==" } }, { "type": "step", "result": "=3u^{2}\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\tan\\left(θ\\right)$$", "result": "=3\\left(\\tan\\left(θ\\right)\\right)^{2}\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYux7TmDJ3fopnHGBxoqO+pYseUrH1bTMvpsf9SfQ9YpVqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKifDI4IHJ+buH5M+0PHLeGB4Be4N1NohQPDRW/BmsNJaPsq1KfCFKwSwA0qmWovUVP9tVYZ/3r82LxPimKUFXjQFStvD8GKDNJJN8qiu0XRcVZAYAvbRKqkRVDaDBHkSBtDwYHkfmHHTze3JXk3LjiJp" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "result": "=\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYs0Vsx3c0nt53Utu5DgHTe3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK00HV/bLwwglXz/2YNx2i4JHggSrmCoGjSdud4CyOvVOgdQN0k+N3NHNAkEIoLrhB631k80nn+ODwI4ID0kCF5+z7zRz81SqRD02HEUE4Wqh" } }, { "type": "step", "result": "=3\\left(\\tan\\left(θ\\right)\\right)^{2}\\sec^{2}\\left(θ\\right)" }, { "type": "step", "primary": "Simplify", "result": "=3\\tan^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=16\\left(4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{3}\\left(θ\\right)+3\\tan^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Simplify $$16\\left(4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{3}\\left(θ\\right)+3\\tan^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right):{\\quad}16\\left(4\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+3\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)$$", "input": "16\\left(4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{3}\\left(θ\\right)+3\\tan^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right)", "result": "=16\\left(4\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+3\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "$$4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{3}\\left(θ\\right)=4\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)$$", "input": "4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{3}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\tan\\left(θ\\right)\\tan^{3}\\left(θ\\right)=\\:\\tan^{1+3}\\left(θ\\right)$$" ], "result": "=4\\sec^{4}\\left(θ\\right)\\tan^{1+3}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+3=4$$", "result": "=4\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7GHerUg4zB8o0EeBO6BKV162cmBc6Js8szDUhC2HxdqjUgkMhq/lXB9qI3pCS5SJMq47vuWedXv2WUg94ER8IwVADhNIEz39BpXewdhMCBb8aab3UY3H06dpMJimNIyD5kuHx4ulZqfxZHIsiQUpV1nyo9elyUFFpcPZW02cAt7EdWYTaqPpaDtAHPW/SgMNicUwY1HWd9EJWOCb4O+kauAND0fAz408eCOOnBAkZ69Akt3WiGR7ZaCaXvz77bMjS" } }, { "type": "interim", "title": "$$3\\tan^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)=3\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)$$", "input": "3\\tan^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)=\\:\\sec^{2+4}\\left(θ\\right)$$" ], "result": "=3\\tan^{2}\\left(θ\\right)\\sec^{2+4}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+4=6$$", "result": "=3\\tan^{2}\\left(θ\\right)\\sec^{6}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7xud7SC9MT1MQdbTGBqrMw+H0qCTik4j9uS7jTF7ng2/8DCHl2/1cUFH2kQZvGLuVzRqDxPUzBN6vjj5oJL9kUEngkySEzEl6diiSTBzPnLnWxcLMxZwB9cGBLGbwbyjTrqr1t4QBx+t+Vf7xBGCGrW9B4jZ4XSgntIkqjxqF2Bvh9Kgk4pOI/bku40xe54Nv6RICywOwLOWusKkGpuhUtYx6/teVLREoXphSPTLZ2MbET1Ehq4mhUZ9DWVx2az0M" } }, { "type": "step", "result": "=16\\left(3\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+4\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7C43rUQmr5KAYI3UGEng6yRiMJmjxigW8LortZKvrqEYERpCXXFEE4hLCZfTeroF6V7JvkhZ2EFhPsvL8iYbkgPpadcGORE5bHX37Io3viWolp/MB1OadflBmLQoq5Pmlo5FYteSPKwXny4uCMrdsKwS9KcmQjztCcdjD5gRjJOz1a03PQ0UrDYBem+/aln9X84iuRh0pbdDnwSddoxT7uh96fmgcT2t23FrnniYZ1CmBBTEk/JQ2cZ9WKuRzClU7UxxUEaCmP0MbiSArdWRj6iHQ+8WidKe5mLtnoi0iGZhpK7ceo/th4B1fP0TRnZr9A1c/UDX6QxYlZyg/KYfMkDSei4QZxxrazp/cx7cYZvYA0hWLB/nt1m4ggrh+mxlbvzIPeEtDfcHv/z8uls8Teg==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)=14\\left(6\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{8}\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=14\\frac{d}{dθ}\\left(\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the Product Rule: $$\\left(f{\\cdot}g\\right)'=f'{\\cdot}g+f{\\cdot}g'$$", "secondary": [ "$$f=\\sec^{6}\\left(θ\\right),\\:g=\\tan\\left(θ\\right)$$" ], "result": "=14\\left(\\frac{d}{dθ}\\left(\\sec^{6}\\left(θ\\right)\\right)\\tan\\left(θ\\right)+\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)\\sec^{6}\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Product%20Rule", "practiceTopic": "Product Rule" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec^{6}\\left(θ\\right)\\right)=6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{6}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}6\\left(\\sec\\left(θ\\right)\\right)^{5}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{6}\\left(θ\\right)\\right)", "result": "=6\\left(\\sec\\left(θ\\right)\\right)^{5}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{6},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{6}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{6}\\right)=6u^{5}$$", "input": "\\frac{d}{du}\\left(u^{6}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=6u^{6-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=6u^{5}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYgOK76JOSNqGa1B1wU6ozWWk3hxk9aCfAWodBRxXgUexqSoDkuZ4Ab1St3bM8oqKcP8//6/nV5O4fb8Xgwi7mapNFxUvwBeni+JEIFAdbegzHfwjUTnSu4cWqw+8m79+NA==" } }, { "type": "step", "result": "=6u^{5}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=6\\left(\\sec\\left(θ\\right)\\right)^{5}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgBaJTGF5uPwSMzxoSyJNVZsqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKieMSe35zYqdvyntnEYxaQ88jJq7JZFtKgvxzzzuaqTGrebfnOah8gAvLMKOeNMctC5tVYZ/3r82LxPimKUFXjQFStvD8GKDNJJN8qiu0XRcVZAYAvbRKqkRVDaDBHkSBtDwYHkfmHHTze3JXk3LjiJp" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=6\\left(\\sec\\left(θ\\right)\\right)^{5}\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$6\\sec^{5}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "6\\sec^{5}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{5}\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{5+1}\\left(θ\\right)$$" ], "result": "=6\\sec^{5+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$5+1=6$$", "result": "=6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7QMJmgA2Sx2hYpGS5j/h1NLPmAPxuc/RFHWXkB8A+mnqLeJKmeakO1+Q3aSIldG417JiujyaxykLR7Tv0Nr+wOGprXgPFMmP3i5vY2oD+e4YslHPSvPf5UfEs5T+IFZ4XHjb2+5NLFZrsH9fcPWg/TRiIlnZT/wCWlhB0uSVNtvx/uR/EZqc6EbI3tnxSI5001Zrc/SYRIR8JuxYpD+PJ+Q==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "result": "=\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYs0Vsx3c0nt53Utu5DgHTe3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK00HV/bLwwglXz/2YNx2i4JHggSrmCoGjSdud4CyOvVOgdQN0k+N3NHNAkEIoLrhB631k80nn+ODwI4ID0kCF5+z7zRz81SqRD02HEUE4Wqh" } }, { "type": "step", "result": "=14\\left(6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{6}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Simplify $$14\\left(6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{6}\\left(θ\\right)\\right):{\\quad}14\\left(6\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{8}\\left(θ\\right)\\right)$$", "input": "14\\left(6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{6}\\left(θ\\right)\\right)", "result": "=14\\left(6\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{8}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "$$6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)=6\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)$$", "input": "6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\tan\\left(θ\\right)\\tan\\left(θ\\right)=\\:\\tan^{1+1}\\left(θ\\right)$$" ], "result": "=6\\sec^{6}\\left(θ\\right)\\tan^{1+1}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=6\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7H4jwT8QhunkiDAGNb+RcvaxUZgt6FTmFSHnRsotkqemLeJKmeakO1+Q3aSIldG417JiujyaxykLR7Tv0Nr+wOGprXgPFMmP3i5vY2oD+e4ZALhShbMGZIZsZWTO721Hsn1G20l7DmbyzInDIeLRU/ZYojigZ8i1ymB3KwWBswnjgWFmp7EeEDyNH2q8Tm1k1jHr+15UtEShemFI9MtnYxsRPUSGriaFRn0NZXHZrPQw=" } }, { "type": "interim", "title": "$$\\sec^{2}\\left(θ\\right)\\sec^{6}\\left(θ\\right)=\\sec^{8}\\left(θ\\right)$$", "input": "\\sec^{2}\\left(θ\\right)\\sec^{6}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{2}\\left(θ\\right)\\sec^{6}\\left(θ\\right)=\\:\\sec^{2+6}\\left(θ\\right)$$" ], "result": "=\\sec^{2+6}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+6=8$$", "result": "=\\sec^{8}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7AtrsReczPdUqIdn5/Ev5mH+O3FtTTH45VMxFH8DJ4BEgJ/ZZA32ZInFBpDtxBfiKn3Jyj2VJgBojKQHjINpkkI6d/oVrekw7T/Cky7x1KxcO/U6qyRZonn1WPQ/21pz3HK+miBBOGkRkZzIN3nn5xFhBmsSvLw1yh3wbSiSoUDgkt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "=14\\left(\\sec^{8}\\left(θ\\right)+6\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7nO0mFYRqkI6v7Sy9VTWwjRiMJmjxigW8LortZKvrqEbWPF6oWuXggupsFvXrooIqHK+miBBOGkRkZzIN3nn5xNMvHyY50dhXPFfrjcmooUiPqGSxzi9gRanHsLmpQgKwsfoApmSO56Ak/1ybzTsFLmprXgPFMmP3i5vY2oD+e4b4ZAjrJM4mwvgfYovsxoopgQUxJPyUNnGfVirkcwpVO2TUlQSCHNJw/4wJ8m9ormYqoLNuWbSjE1JT+hrPzX/RhYbzXwn7iok10o2w+NJLRiZIb9CFuBJdPyi+MBoLCMeBkcS/lgaNKtB6zcW1E3BB" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=24\\left(16\\left(4\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+3\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)+14\\left(6\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{8}\\left(θ\\right)\\right)\\right)" }, { "type": "interim", "title": "Expand $$16\\left(4\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+3\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)+14\\left(6\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{8}\\left(θ\\right)\\right):{\\quad}64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+132\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+14\\sec^{8}\\left(θ\\right)$$", "input": "16\\left(4\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+3\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)+14\\left(6\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{8}\\left(θ\\right)\\right)", "result": "=24\\left(64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+132\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+14\\sec^{8}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Expand $$16\\left(4\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+3\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right):{\\quad}64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+48\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)$$", "input": "16\\left(4\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+3\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)", "result": "=64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+48\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+14\\left(6\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{8}\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b+c\\right)=ab+ac$$", "secondary": [ "$$a=16,\\:b=4\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right),\\:c=3\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)$$" ], "result": "=16\\cdot\\:4\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+16\\cdot\\:3\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "interim", "title": "Simplify $$16\\cdot\\:4\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+16\\cdot\\:3\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right):{\\quad}64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+48\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)$$", "input": "16\\cdot\\:4\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+16\\cdot\\:3\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)", "result": "=64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+48\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$16\\cdot\\:4=64$$", "result": "=64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+16\\cdot\\:3\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)" }, { "type": "step", "primary": "Multiply the numbers: $$16\\cdot\\:3=48$$", "result": "=64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+48\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7C43rUQmr5KAYI3UGEng6yV6kSs12eASfWmxynRkYZ3imWJLfEdyBmZh9L6HIlMnFSPKgibEOx6xAH14rsVQdDd6GQqufR6tr2vPxOUv7H+/wV+6wolBQOY9LPugwYJ6C3tKI9zGyi4xCTaccmboyHVWGy3JBiLa3A/i0ciCokDrRsdR53zGMNMouUmBvz53xupxJf7boNv1S9Hc0DXJXvt7UD16RqFQha281as3i/3gIhVb1SjznyQ5krYRiFqNmoU0M/hfYOSS/FiVCoXaNrXxGVgveSDjBoBB0CLM8z96ArlFiGY8qAdw44hDradHvvzIPeEtDfcHv/z8uls8Teg==" } }, { "type": "interim", "title": "Expand $$14\\left(6\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{8}\\left(θ\\right)\\right):{\\quad}84\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+14\\sec^{8}\\left(θ\\right)$$", "input": "14\\left(6\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{8}\\left(θ\\right)\\right)", "result": "=64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+48\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+84\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+14\\sec^{8}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b+c\\right)=ab+ac$$", "secondary": [ "$$a=14,\\:b=6\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right),\\:c=\\sec^{8}\\left(θ\\right)$$" ], "result": "=14\\cdot\\:6\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+14\\sec^{8}\\left(θ\\right)", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "primary": "Multiply the numbers: $$14\\cdot\\:6=84$$", "result": "=84\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+14\\sec^{8}\\left(θ\\right)" } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7nO0mFYRqkI6v7Sy9VTWwjSvxearhctbAPxX3QGBh5kkx0tWj1vW7tIvP14Mwoy6mLTrWWMFI8l4Q07DZ5+hJazPkBMIuMrWSeavgWBoF37pgEq4U4c+VEIUu/cb6SEfLgBHHd6NlsvdNQCpoDejgKWxRFK5IwscCsNiX9YX0G6d6pfF1z6umzUJTJvt+ojYZ00PxkCV/xFqV0mmm+Nwrth1Y04xtNK2BCM2tE3PxmDme5tn70JIT2jKyMskERPvitGNYJGuT8kFILfu1szjq+A==" } }, { "type": "step", "primary": "Add similar elements: $$48\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+84\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)=132\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)$$", "result": "=64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+132\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+14\\sec^{8}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7C43rUQmr5KAYI3UGEng6yV6kSs12eASfWmxynRkYZ3imWJLfEdyBmZh9L6HIlMnFquwP5S7dVU2hOl0jx4UuGyqgs25ZtKMTUlP6Gs/Nf9EOOEJ8eYI1Z69ODewOg6d0GN3NaV3za5y0fd8sDMMqbt13jtrSFDx+UNsawjlOjV0T07YNGlOvsbUamOsS8Bedjha9s6alqYoo6ur9CV5YJR67hL3SLnd3F8UQKch9b3y7ssmRiNW3dHLSNd768UtCbqAWk+sYoFfdaspteys3OK6q9beEAcfrflX+8QRghq2CcvL2jSzw1VlRcGX4ka6qUYDxHhHJhPwBywaMxzqJZlGGuUmWLwacJdjIdVxl7S4IFLNGVGiL7NHRmdqGkv+kzFx+puj8P4X++28hfOTEK4y85Xs96ub4hnNPpGMCnfiCWjNEoRZwVGhuwKG+ZGEU32vN1Q0V3hnPq2BOOEcCJA==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=-8\\left(-4\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)+12\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)\\right)+24\\left(64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+132\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+14\\sec^{8}\\left(θ\\right)\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=-8\\left(-4\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)+12\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)\\right)+24\\left(64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+132\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+14\\sec^{8}\\left(θ\\right)\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "Evaluate $$-8\\left(-4\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)+12\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)\\right)+24\\left(64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+132\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+14\\sec^{8}\\left(θ\\right)\\right)\\:$$at point $$θ=0:{\\quad}272$$", "result": "=272", "steps": [ { "type": "step", "primary": "Take the point $$θ=0$$ and plug it into $$-8\\left(-4\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)+12\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)\\right)+24\\left(64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+132\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+14\\sec^{8}\\left(θ\\right)\\right)$$", "result": "=-8\\left(-4\\left(-2\\sec^{2}\\left(0\\right)+3\\sec^{4}\\left(0\\right)\\right)+12\\left(4\\sec^{4}\\left(0\\right)\\tan^{2}\\left(0\\right)+\\sec^{6}\\left(0\\right)\\right)\\right)+24\\left(64\\sec^{4}\\left(0\\right)\\tan^{4}\\left(0\\right)+132\\sec^{6}\\left(0\\right)\\tan^{2}\\left(0\\right)+14\\sec^{8}\\left(0\\right)\\right)" }, { "type": "interim", "title": "$$8\\left(-4\\left(-2\\sec^{2}\\left(0\\right)+3\\sec^{4}\\left(0\\right)\\right)+12\\left(4\\sec^{4}\\left(0\\right)\\tan^{2}\\left(0\\right)+\\sec^{6}\\left(0\\right)\\right)\\right)=64$$", "input": "8\\left(-4\\left(-2\\sec^{2}\\left(0\\right)+3\\sec^{4}\\left(0\\right)\\right)+12\\left(4\\sec^{4}\\left(0\\right)\\tan^{2}\\left(0\\right)+\\sec^{6}\\left(0\\right)\\right)\\right)", "steps": [ { "type": "interim", "title": "$$4\\left(-2\\sec^{2}\\left(0\\right)+3\\sec^{4}\\left(0\\right)\\right)=4$$", "input": "4\\left(-2\\sec^{2}\\left(0\\right)+3\\sec^{4}\\left(0\\right)\\right)", "steps": [ { "type": "interim", "title": "$$2\\sec^{2}\\left(0\\right)=2$$", "input": "2\\sec^{2}\\left(0\\right)", "steps": [ { "type": "interim", "title": "$$\\sec^{2}\\left(0\\right)=1$$", "input": "\\sec^{2}\\left(0\\right)", "steps": [ { "type": "interim", "title": "Simplify $$\\sec\\left(0\\right):{\\quad}1$$", "input": "\\sec\\left(0\\right)", "result": "=1^{2}", "steps": [ { "type": "step", "primary": "Use the following trivial identity:$${\\quad}\\sec\\left(0\\right)=1$$", "secondary": [ "$$\\sec\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sec(x)&x&\\sec(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{2\\sqrt{3}}{3}&\\frac{7π}{6}&-\\frac{2\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&\\sqrt{2}&\\frac{5π}{4}&-\\sqrt{2}\\\\\\hline \\frac{π}{3}&2&\\frac{4π}{3}&-2\\\\\\hline \\frac{π}{2}&\\mathrm{Undefined}&\\frac{3π}{2}&\\mathrm{Undefined}\\\\\\hline \\frac{2π}{3}&-2&\\frac{5π}{3}&2\\\\\\hline \\frac{3π}{4}&-\\sqrt{2}&\\frac{7π}{4}&\\sqrt{2}\\\\\\hline \\frac{5π}{6}&-\\frac{2\\sqrt{3}}{3}&\\frac{11π}{6}&\\frac{2\\sqrt{3}}{3}\\\\\\hline \\end{array}$$" ], "result": "=1" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7y1xaZeQTSzp609voOcYk6VXTSum/z5kLpMzXS1UJIexiYFOaxxQqg3o8CNKkTm2qW+Sc57RV820557NJqaC71BfVjTo90KMWspnEdE4CKjE=" } }, { "type": "step", "result": "=2\\cdot\\:1" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=2" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7WiIIsb80YMnC/PcLPmbAry061ljBSPJeENOw2efoSWt8rNweSBKaNqhWM5iGGSVMdOMUzZRHPIlp1R8T2pkp20fQYskTwd8nVLo5risF1/I=" } }, { "type": "interim", "title": "$$3\\sec^{4}\\left(0\\right)=3$$", "input": "3\\sec^{4}\\left(0\\right)", "steps": [ { "type": "interim", "title": "$$\\sec^{4}\\left(0\\right)=1$$", "input": "\\sec^{4}\\left(0\\right)", "steps": [ { "type": "interim", "title": "Simplify $$\\sec\\left(0\\right):{\\quad}1$$", "input": "\\sec\\left(0\\right)", "result": "=1^{4}", "steps": [ { "type": "step", "primary": "Use the following trivial identity:$${\\quad}\\sec\\left(0\\right)=1$$", "secondary": [ "$$\\sec\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sec(x)&x&\\sec(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{2\\sqrt{3}}{3}&\\frac{7π}{6}&-\\frac{2\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&\\sqrt{2}&\\frac{5π}{4}&-\\sqrt{2}\\\\\\hline \\frac{π}{3}&2&\\frac{4π}{3}&-2\\\\\\hline \\frac{π}{2}&\\mathrm{Undefined}&\\frac{3π}{2}&\\mathrm{Undefined}\\\\\\hline \\frac{2π}{3}&-2&\\frac{5π}{3}&2\\\\\\hline \\frac{3π}{4}&-\\sqrt{2}&\\frac{7π}{4}&\\sqrt{2}\\\\\\hline \\frac{5π}{6}&-\\frac{2\\sqrt{3}}{3}&\\frac{11π}{6}&\\frac{2\\sqrt{3}}{3}\\\\\\hline \\end{array}$$" ], "result": "=1" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7TnbPBMsEd78kmmwfYDHqNVXTSum/z5kLpMzXS1UJIexiYFOaxxQqg3o8CNKkTm2qW+Sc57RV820557NJqaC71Ka6Ma3rg33eDIhfX/iuVnM=" } }, { "type": "step", "result": "=3\\cdot\\:1" }, { "type": "step", "primary": "Multiply the numbers: $$3\\cdot\\:1=3$$", "result": "=3" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/DCWessgznXvbIt+KUH6Sy061ljBSPJeENOw2efoSWuWTYGorfWzSEgF1o3crWNtABnhaHvVHSBGAVNgbL75+DAm81Br6rrrU7LVgwPHPmM=" } }, { "type": "step", "result": "=4\\left(3-2\\right)" }, { "type": "step", "primary": "Add/Subtract the numbers: $$-2+3=1$$", "result": "=4\\cdot\\:1" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:1=4$$", "result": "=4" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7SGVSB79Ogsx4oIuNaHoM5zOWn8zcWOZXWMkYneaKW/PTLx8mOdHYVzxX643JqKFIlFYe8lM/WYPz8s80IajfKB0R3c0uYNhyRWpx951US4jV06frS5K/+G+6QAS0YeGxwllEClEEu5y+6CgQm6mC37CI2sSeA74029n2yo277ZU=" } }, { "type": "interim", "title": "$$12\\left(4\\sec^{4}\\left(0\\right)\\tan^{2}\\left(0\\right)+\\sec^{6}\\left(0\\right)\\right)=12$$", "input": "12\\left(4\\sec^{4}\\left(0\\right)\\tan^{2}\\left(0\\right)+\\sec^{6}\\left(0\\right)\\right)", "steps": [ { "type": "interim", "title": "$$4\\sec^{4}\\left(0\\right)\\tan^{2}\\left(0\\right)=0$$", "input": "4\\sec^{4}\\left(0\\right)\\tan^{2}\\left(0\\right)", "steps": [ { "type": "interim", "title": "$$\\sec^{4}\\left(0\\right)=1$$", "input": "\\sec^{4}\\left(0\\right)", "steps": [ { "type": "interim", "title": "Simplify $$\\sec\\left(0\\right):{\\quad}1$$", "input": "\\sec\\left(0\\right)", "result": "=1^{4}", "steps": [ { "type": "step", "primary": "Use the following trivial identity:$${\\quad}\\sec\\left(0\\right)=1$$", "secondary": [ "$$\\sec\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sec(x)&x&\\sec(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{2\\sqrt{3}}{3}&\\frac{7π}{6}&-\\frac{2\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&\\sqrt{2}&\\frac{5π}{4}&-\\sqrt{2}\\\\\\hline \\frac{π}{3}&2&\\frac{4π}{3}&-2\\\\\\hline \\frac{π}{2}&\\mathrm{Undefined}&\\frac{3π}{2}&\\mathrm{Undefined}\\\\\\hline \\frac{2π}{3}&-2&\\frac{5π}{3}&2\\\\\\hline \\frac{3π}{4}&-\\sqrt{2}&\\frac{7π}{4}&\\sqrt{2}\\\\\\hline \\frac{5π}{6}&-\\frac{2\\sqrt{3}}{3}&\\frac{11π}{6}&\\frac{2\\sqrt{3}}{3}\\\\\\hline \\end{array}$$" ], "result": "=1" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7TnbPBMsEd78kmmwfYDHqNVXTSum/z5kLpMzXS1UJIexiYFOaxxQqg3o8CNKkTm2qW+Sc57RV820557NJqaC71Ka6Ma3rg33eDIhfX/iuVnM=" } }, { "type": "step", "result": "=4\\cdot\\:1\\cdot\\:\\tan^{2}\\left(0\\right)" }, { "type": "interim", "title": "$$\\tan^{2}\\left(0\\right)=0$$", "input": "\\tan^{2}\\left(0\\right)", "steps": [ { "type": "interim", "title": "Simplify $$\\tan\\left(0\\right):{\\quad}0$$", "input": "\\tan\\left(0\\right)", "result": "=0^{2}", "steps": [ { "type": "step", "primary": "Use the following trivial identity:$${\\quad}\\tan\\left(0\\right)=0$$", "secondary": [ "$$\\tan\\left(x\\right)$$ periodicity table with $$πn$$ cycle:<br/>$$\\begin{array}{|c|c|}\\hline x&\\tan(x)\\\\\\hline 0&0\\\\\\hline \\frac{π}{6}&\\frac{\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&1\\\\\\hline \\frac{π}{3}&\\sqrt{3}\\\\\\hline \\frac{π}{2}&\\pm\\infty\\\\\\hline \\frac{2π}{3}&-\\sqrt{3}\\\\\\hline \\frac{3π}{4}&-1\\\\\\hline \\frac{5π}{6}&-\\frac{\\sqrt{3}}{3}\\\\\\hline &\\\\\\hline \\end{array}$$" ], "result": "=0" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Apply rule $$0^{a}=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7EFsToMX1NPjPuXVSALucCFXTSum/z5kLpMzXS1UJIeyqjfZqhkrOoAz8sRb7wXZWPGHkI0MFUiTEwg/g9Re5WBSGyzDy0veA23V3wI6HDZ0=" } }, { "type": "step", "result": "=4\\cdot\\:1\\cdot\\:0" }, { "type": "step", "primary": "Apply rule $$0\\cdot\\:a=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s78lx/rXvIfCfIw4X8Nv9JwGpb6Zt5uD6Xe9ICIhSFAqbdd47a0hQ8flDbGsI5To1dTbAOxT8wOTlsw5gGf+Hdr8r4WVs4CtopdU5PWLTw1eRWrxRA8eaaGXYlDOSEM378ENKOm3B8+xM3yGSItlc9gA==" } }, { "type": "interim", "title": "$$\\sec^{6}\\left(0\\right)=1$$", "input": "\\sec^{6}\\left(0\\right)", "steps": [ { "type": "interim", "title": "Simplify $$\\sec\\left(0\\right):{\\quad}1$$", "input": "\\sec\\left(0\\right)", "result": "=1^{6}", "steps": [ { "type": "step", "primary": "Use the following trivial identity:$${\\quad}\\sec\\left(0\\right)=1$$", "secondary": [ "$$\\sec\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sec(x)&x&\\sec(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{2\\sqrt{3}}{3}&\\frac{7π}{6}&-\\frac{2\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&\\sqrt{2}&\\frac{5π}{4}&-\\sqrt{2}\\\\\\hline \\frac{π}{3}&2&\\frac{4π}{3}&-2\\\\\\hline \\frac{π}{2}&\\mathrm{Undefined}&\\frac{3π}{2}&\\mathrm{Undefined}\\\\\\hline \\frac{2π}{3}&-2&\\frac{5π}{3}&2\\\\\\hline \\frac{3π}{4}&-\\sqrt{2}&\\frac{7π}{4}&\\sqrt{2}\\\\\\hline \\frac{5π}{6}&-\\frac{2\\sqrt{3}}{3}&\\frac{11π}{6}&\\frac{2\\sqrt{3}}{3}\\\\\\hline \\end{array}$$" ], "result": "=1" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7WcSYqbVPn/qsbXXyqHdMbFXTSum/z5kLpMzXS1UJIexiYFOaxxQqg3o8CNKkTm2qW+Sc57RV820557NJqaC71Idwe88IvH72vjqcga7gyBg=" } }, { "type": "step", "result": "=12\\left(0+1\\right)" }, { "type": "step", "primary": "Add the numbers: $$0+1=1$$", "result": "=12\\cdot\\:1" }, { "type": "step", "primary": "Multiply the numbers: $$12\\cdot\\:1=12$$", "result": "=12" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7lv52tOJZUIL6vqIybdOpTv6k/0Ej05jaZlvKuZ7BchqkNRU526e9DUedYVjkPVRIICf2WQN9mSJxQaQ7cQX4iveB/AqBp60vexPOzIrQ2kDh6XJOU+ADur+UsyVO4as2WfKjICzStUBkrvQwB+qiDavP5dLtToEw2Zpkikf00/ZOjeRcPaACRL5La6ZwWTik" } }, { "type": "step", "result": "=8\\left(12-4\\right)" }, { "type": "step", "primary": "Add/Subtract the numbers: $$-4+12=8$$", "result": "=8\\cdot\\:8" }, { "type": "step", "primary": "Multiply the numbers: $$8\\cdot\\:8=64$$", "result": "=64" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7pEUJy+CtYVEiKzNsPSlaeg9zA7z3N/kwupn+iB41A+xqlU3QAFTFK+yGmPKWoDuzazhA0+a1wf9G1fiOsZuQNzomUw5kYyCrSM1rC9ftEoUAlilG71elit3w1IBbYN0PWhgO/ZWG/9BpGzUXX1uMCY08PkFfwa+ZiJNMY6gHh8PYY6TiTc30dYTBcnzGf4+8M5afzNxY5ldYyRid5opb84FAqMTCfEIxPAs+HdURhAz+pP9BI9OY2mZbyrmewXIa1Bn8tv67SHuBTDtzJdy34bCI2sSeA74029n2yo277ZU=" } }, { "type": "interim", "title": "$$24\\left(64\\sec^{4}\\left(0\\right)\\tan^{4}\\left(0\\right)+132\\sec^{6}\\left(0\\right)\\tan^{2}\\left(0\\right)+14\\sec^{8}\\left(0\\right)\\right)=336$$", "input": "24\\left(64\\sec^{4}\\left(0\\right)\\tan^{4}\\left(0\\right)+132\\sec^{6}\\left(0\\right)\\tan^{2}\\left(0\\right)+14\\sec^{8}\\left(0\\right)\\right)", "steps": [ { "type": "interim", "title": "$$64\\sec^{4}\\left(0\\right)\\tan^{4}\\left(0\\right)=0$$", "input": "64\\sec^{4}\\left(0\\right)\\tan^{4}\\left(0\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{m}b^{m}=\\left(ab\\right)^{m}$$", "secondary": [ "$$\\sec^{4}\\left(0\\right)\\tan^{4}\\left(0\\right)=\\left(\\sec\\left(0\\right)\\tan\\left(0\\right)\\right)^{4}$$" ], "result": "=64\\left(\\sec\\left(0\\right)\\tan\\left(0\\right)\\right)^{4}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "$$\\sec\\left(0\\right)\\tan\\left(0\\right)=0$$", "input": "\\sec\\left(0\\right)\\tan\\left(0\\right)", "steps": [ { "type": "interim", "title": "Simplify $$\\sec\\left(0\\right):{\\quad}1$$", "input": "\\sec\\left(0\\right)", "result": "=1\\cdot\\:\\tan\\left(0\\right)", "steps": [ { "type": "step", "primary": "Use the following trivial identity:$${\\quad}\\sec\\left(0\\right)=1$$", "secondary": [ "$$\\sec\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sec(x)&x&\\sec(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{2\\sqrt{3}}{3}&\\frac{7π}{6}&-\\frac{2\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&\\sqrt{2}&\\frac{5π}{4}&-\\sqrt{2}\\\\\\hline \\frac{π}{3}&2&\\frac{4π}{3}&-2\\\\\\hline \\frac{π}{2}&\\mathrm{Undefined}&\\frac{3π}{2}&\\mathrm{Undefined}\\\\\\hline \\frac{2π}{3}&-2&\\frac{5π}{3}&2\\\\\\hline \\frac{3π}{4}&-\\sqrt{2}&\\frac{7π}{4}&\\sqrt{2}\\\\\\hline \\frac{5π}{6}&-\\frac{2\\sqrt{3}}{3}&\\frac{11π}{6}&\\frac{2\\sqrt{3}}{3}\\\\\\hline \\end{array}$$" ], "result": "=1" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "interim", "title": "Simplify $$\\tan\\left(0\\right):{\\quad}0$$", "input": "\\tan\\left(0\\right)", "result": "=1\\cdot\\:0", "steps": [ { "type": "step", "primary": "Use the following trivial identity:$${\\quad}\\tan\\left(0\\right)=0$$", "secondary": [ "$$\\tan\\left(x\\right)$$ periodicity table with $$πn$$ cycle:<br/>$$\\begin{array}{|c|c|}\\hline x&\\tan(x)\\\\\\hline 0&0\\\\\\hline \\frac{π}{6}&\\frac{\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&1\\\\\\hline \\frac{π}{3}&\\sqrt{3}\\\\\\hline \\frac{π}{2}&\\pm\\infty\\\\\\hline \\frac{2π}{3}&-\\sqrt{3}\\\\\\hline \\frac{3π}{4}&-1\\\\\\hline \\frac{5π}{6}&-\\frac{\\sqrt{3}}{3}\\\\\\hline &\\\\\\hline \\end{array}$$" ], "result": "=0" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Apply rule $$0\\cdot\\:a=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7wZhLm0FifI5RV/wdUY/V+xs4UrvXrnDv0XDlfU7q7ckE5aqGN/sLZfeoFZRwtGLqP8vQyhiD4JSfqjIvcQ7tinNBgJ91kDwEQa4/+Opm+A8QRYlc0DxGPn21AQgIWYWg" } }, { "type": "step", "result": "=0^{4}\\cdot\\:64" }, { "type": "step", "primary": "Apply rule $$0^{a}=0$$", "result": "=64\\cdot\\:0" }, { "type": "step", "primary": "Apply rule $$0\\cdot\\:a=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7jCpfsFjEU27xNecKdQGhyN+vnf11FNHf3iNt9M+J8Z8gJ/ZZA32ZInFBpDtxBfiKf7LqB9CcyvYCWDsGseX09kk96RFLStITDoCO0hmLjEzqP13v7CCUzv+TjUxfUDRqCFX7kPfEkIuYfqIFW07ybg==" } }, { "type": "interim", "title": "$$132\\sec^{6}\\left(0\\right)\\tan^{2}\\left(0\\right)=0$$", "input": "132\\sec^{6}\\left(0\\right)\\tan^{2}\\left(0\\right)", "steps": [ { "type": "interim", "title": "$$\\sec^{6}\\left(0\\right)=1$$", "input": "\\sec^{6}\\left(0\\right)", "steps": [ { "type": "interim", "title": "Simplify $$\\sec\\left(0\\right):{\\quad}1$$", "input": "\\sec\\left(0\\right)", "result": "=1^{6}", "steps": [ { "type": "step", "primary": "Use the following trivial identity:$${\\quad}\\sec\\left(0\\right)=1$$", "secondary": [ "$$\\sec\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sec(x)&x&\\sec(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{2\\sqrt{3}}{3}&\\frac{7π}{6}&-\\frac{2\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&\\sqrt{2}&\\frac{5π}{4}&-\\sqrt{2}\\\\\\hline \\frac{π}{3}&2&\\frac{4π}{3}&-2\\\\\\hline \\frac{π}{2}&\\mathrm{Undefined}&\\frac{3π}{2}&\\mathrm{Undefined}\\\\\\hline \\frac{2π}{3}&-2&\\frac{5π}{3}&2\\\\\\hline \\frac{3π}{4}&-\\sqrt{2}&\\frac{7π}{4}&\\sqrt{2}\\\\\\hline \\frac{5π}{6}&-\\frac{2\\sqrt{3}}{3}&\\frac{11π}{6}&\\frac{2\\sqrt{3}}{3}\\\\\\hline \\end{array}$$" ], "result": "=1" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7WcSYqbVPn/qsbXXyqHdMbFXTSum/z5kLpMzXS1UJIexiYFOaxxQqg3o8CNKkTm2qW+Sc57RV820557NJqaC71Idwe88IvH72vjqcga7gyBg=" } }, { "type": "step", "result": "=132\\cdot\\:1\\cdot\\:\\tan^{2}\\left(0\\right)" }, { "type": "interim", "title": "$$\\tan^{2}\\left(0\\right)=0$$", "input": "\\tan^{2}\\left(0\\right)", "steps": [ { "type": "interim", "title": "Simplify $$\\tan\\left(0\\right):{\\quad}0$$", "input": "\\tan\\left(0\\right)", "result": "=0^{2}", "steps": [ { "type": "step", "primary": "Use the following trivial identity:$${\\quad}\\tan\\left(0\\right)=0$$", "secondary": [ "$$\\tan\\left(x\\right)$$ periodicity table with $$πn$$ cycle:<br/>$$\\begin{array}{|c|c|}\\hline x&\\tan(x)\\\\\\hline 0&0\\\\\\hline \\frac{π}{6}&\\frac{\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&1\\\\\\hline \\frac{π}{3}&\\sqrt{3}\\\\\\hline \\frac{π}{2}&\\pm\\infty\\\\\\hline \\frac{2π}{3}&-\\sqrt{3}\\\\\\hline \\frac{3π}{4}&-1\\\\\\hline \\frac{5π}{6}&-\\frac{\\sqrt{3}}{3}\\\\\\hline &\\\\\\hline \\end{array}$$" ], "result": "=0" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Apply rule $$0^{a}=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7EFsToMX1NPjPuXVSALucCFXTSum/z5kLpMzXS1UJIeyqjfZqhkrOoAz8sRb7wXZWPGHkI0MFUiTEwg/g9Re5WBSGyzDy0veA23V3wI6HDZ0=" } }, { "type": "step", "result": "=132\\cdot\\:1\\cdot\\:0" }, { "type": "step", "primary": "Apply rule $$0\\cdot\\:a=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7hDHcMpcibER9XzN1Yg7CFW7J22bskQyWoBdfBnHASMvehkKrn0era9rz8TlL+x/vZuJKdCFsPJy1+5gBMEc9dsdkCcWWh42PXyyVg0WHsjHh/R/PNQmbxXqoSPkeiOuEYAQCAfHsRO0+u9TOmQBHnA==" } }, { "type": "interim", "title": "$$14\\sec^{8}\\left(0\\right)=14$$", "input": "14\\sec^{8}\\left(0\\right)", "steps": [ { "type": "interim", "title": "$$\\sec^{8}\\left(0\\right)=1$$", "input": "\\sec^{8}\\left(0\\right)", "steps": [ { "type": "interim", "title": "Simplify $$\\sec\\left(0\\right):{\\quad}1$$", "input": "\\sec\\left(0\\right)", "result": "=1^{8}", "steps": [ { "type": "step", "primary": "Use the following trivial identity:$${\\quad}\\sec\\left(0\\right)=1$$", "secondary": [ "$$\\sec\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sec(x)&x&\\sec(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{2\\sqrt{3}}{3}&\\frac{7π}{6}&-\\frac{2\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&\\sqrt{2}&\\frac{5π}{4}&-\\sqrt{2}\\\\\\hline \\frac{π}{3}&2&\\frac{4π}{3}&-2\\\\\\hline \\frac{π}{2}&\\mathrm{Undefined}&\\frac{3π}{2}&\\mathrm{Undefined}\\\\\\hline \\frac{2π}{3}&-2&\\frac{5π}{3}&2\\\\\\hline \\frac{3π}{4}&-\\sqrt{2}&\\frac{7π}{4}&\\sqrt{2}\\\\\\hline \\frac{5π}{6}&-\\frac{2\\sqrt{3}}{3}&\\frac{11π}{6}&\\frac{2\\sqrt{3}}{3}\\\\\\hline \\end{array}$$" ], "result": "=1" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7p6IxEfKejassVlEaaYgXqFXTSum/z5kLpMzXS1UJIexiYFOaxxQqg3o8CNKkTm2qW+Sc57RV820557NJqaC71Dd1z7LYeV9QGyBQ1bCNf14=" } }, { "type": "step", "result": "=14\\cdot\\:1" }, { "type": "step", "primary": "Multiply the numbers: $$14\\cdot\\:1=14$$", "result": "=14" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7CKfnUOCj05zZQSfHGtbxZgCWKUbvV6WK3fDUgFtg3Q9I5fbYu61vDlo8ZYCzwCZ5jFF+Grhte/2UqFkzsPs4p48kfsp1NuZUpmGmGCq3EH6wiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=24\\left(0+0+14\\right)" }, { "type": "step", "primary": "Add the numbers: $$0+0+14=14$$", "result": "=24\\cdot\\:14" }, { "type": "step", "primary": "Multiply the numbers: $$24\\cdot\\:14=336$$", "result": "=336" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s78nQMBHu+prnCCVw5o1J/eVxgK+yPINzyLi5gcEB0VbiOfWofmcg617UzDkH+2fVbyR3NdNPppln3fSgAjl+TBB1Pl0yjHH89T351Qec2oqHdd47a0hQ8flDbGsI5To1d0FZ2CrTVRlVnY8uehUEtpO18KGKXzANuFVXr2WUkL90zq+QS9kblohPScFnBjh/0CTAFHH16SB6RSMGSVmr0OZWBls1W2NWhPslgpbulGoy5I4rnDsAX1ck0KsGYmrx315/xlwUV+z5+nn1kL7hm9w==" } }, { "type": "step", "result": "=-64+336" }, { "type": "step", "primary": "Simplify" }, { "type": "step", "result": "=272" } ], "meta": { "solvingClass": "Solver", "interimType": "Laplace Eval Function At Point Specific 3Eq" } }, { "type": "step", "result": "272" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\frac{d^{8}}{dθ^{8}}\\left(\\tan\\left(θ\\right)\\right)\\left(0\\right){\\quad:\\quad}0$$", "input": "\\frac{d^{8}}{dθ^{8}}\\left(\\tan\\left(θ\\right)\\right)\\left(0\\right)", "steps": [ { "type": "interim", "title": "$$\\frac{d^{8}}{dθ^{8}}\\left(\\tan\\left(θ\\right)\\right)=-8\\left(-4\\left(-4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+12\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)\\right)+24\\left(256\\tan^{5}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+1048\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+376\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)\\right)$$", "input": "\\frac{d^{8}}{dθ^{8}}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "result": "=\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYs0Vsx3c0nt53Utu5DgHTe3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK00HV/bLwwglXz/2YNx2i4JHggSrmCoGjSdud4CyOvVOgdQN0k+N3NHNAkEIoLrhB631k80nn+ODwI4ID0kCF5+z7zRz81SqRD02HEUE4Wqh" } }, { "type": "step", "result": "=\\frac{d^{7}}{dθ^{7}}\\left(\\sec^{2}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{2},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{2}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{2}\\right)=2u$$", "input": "\\frac{d}{du}\\left(u^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2u^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2u", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYr+VZUwBnLdzbS6DZQ+f4s+k3hxk9aCfAWodBRxXgUexMchyqTAJWrzJaDbnNcFsJUeCBKuYKgaNJ253gLI69U79qbCA2QqVmvm3jGRXZ2ppvbGT4j1utMEkCDH25m/vlQ==" } }, { "type": "step", "result": "=2u\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgCSXIThcWH58oYoMPt18lzFqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidZClirlWNFPnHR1rvdyoFLI7IC+oqnwb1T6nPLpiw1eysw2NXPd8jZnr3fcPpqc7ejeh7+jKEzLb7VNCEMF3Z/Maswa5oIagKQFOfrnKm5+Gbb/PxX3awut5KxRxyZf8A=" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{1+1}\\left(θ\\right)$$" ], "result": "=2\\sec^{1+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7aY4rYekMZhUfP6sZ6mfqd6Lsz0lt1XLEr88Kc/cjJo5V00rpv8+ZC6TM10tVCSHspqrCcTwb/I0f8OwMjuU6eBAUa+AHsHd5OGpMiuFITjOuqvW3hAHH635V/vEEYIatHimBRYRqHSWeJkuUPhfTC5LQ0B1JNgsb5SXzbS+SCnAfGB9H25Zc+Ua1z7Kw8kgb4gBJl4WMO1rA0a30/bUYlg==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{d^{6}}{dθ^{6}}\\left(2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\right)=-4\\sec^{2}\\left(θ\\right)+6\\sec^{4}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=2\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the Product Rule: $$\\left(f{\\cdot}g\\right)'=f'{\\cdot}g+f{\\cdot}g'$$", "secondary": [ "$$f=\\sec^{2}\\left(θ\\right),\\:g=\\tan\\left(θ\\right)$$" ], "result": "=2\\left(\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)\\tan\\left(θ\\right)+\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)\\sec^{2}\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Product%20Rule", "practiceTopic": "Product Rule" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{2},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{2}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{2}\\right)=2u$$", "input": "\\frac{d}{du}\\left(u^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2u^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2u", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYr+VZUwBnLdzbS6DZQ+f4s+k3hxk9aCfAWodBRxXgUexMchyqTAJWrzJaDbnNcFsJUeCBKuYKgaNJ253gLI69U79qbCA2QqVmvm3jGRXZ2ppvbGT4j1utMEkCDH25m/vlQ==" } }, { "type": "step", "result": "=2u\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgCSXIThcWH58oYoMPt18lzFqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidZClirlWNFPnHR1rvdyoFLI7IC+oqnwb1T6nPLpiw1eysw2NXPd8jZnr3fcPpqc7ejeh7+jKEzLb7VNCEMF3Z/Maswa5oIagKQFOfrnKm5+Gbb/PxX3awut5KxRxyZf8A=" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{1+1}\\left(θ\\right)$$" ], "result": "=2\\sec^{1+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7aY4rYekMZhUfP6sZ6mfqd6Lsz0lt1XLEr88Kc/cjJo5V00rpv8+ZC6TM10tVCSHspqrCcTwb/I0f8OwMjuU6eBAUa+AHsHd5OGpMiuFITjOuqvW3hAHH635V/vEEYIatHimBRYRqHSWeJkuUPhfTC5LQ0B1JNgsb5SXzbS+SCnAfGB9H25Zc+Ua1z7Kw8kgb4gBJl4WMO1rA0a30/bUYlg==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "result": "=\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYs0Vsx3c0nt53Utu5DgHTe3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK00HV/bLwwglXz/2YNx2i4JHggSrmCoGjSdud4CyOvVOgdQN0k+N3NHNAkEIoLrhB631k80nn+ODwI4ID0kCF5+z7zRz81SqRD02HEUE4Wqh" } }, { "type": "step", "result": "=2\\left(2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Simplify $$2\\left(2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\right):{\\quad}-4\\sec^{2}\\left(θ\\right)+6\\sec^{4}\\left(θ\\right)$$", "input": "2\\left(2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\right)", "result": "=-4\\sec^{2}\\left(θ\\right)+6\\sec^{4}\\left(θ\\right)", "steps": [ { "type": "interim", "title": "$$2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)=2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)$$", "input": "2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\tan\\left(θ\\right)\\tan\\left(θ\\right)=\\:\\tan^{1+1}\\left(θ\\right)$$" ], "result": "=2\\sec^{2}\\left(θ\\right)\\tan^{1+1}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Md1SJKHYLxhOYRRbsLTu+axUZgt6FTmFSHnRsotkqemLeJKmeakO1+Q3aSIldG41s7BwSVdKEYSqn+4fKAQoGk0xmFIl1KTO5aAkhaHvKzBALhShbMGZIZsZWTO721HsWI4dG63F/aSz/PFDvu5FnO3sngYNzcydIedGQjjtM8sBMUhkLMqDD6KHvgZAgKMdRQzIIgXxnWeDhDTk2Lfum8RPUSGriaFRn0NZXHZrPQw=" } }, { "type": "interim", "title": "$$\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)=\\sec^{4}\\left(θ\\right)$$", "input": "\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)=\\:\\sec^{2+2}\\left(θ\\right)$$" ], "result": "=\\sec^{2+2}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+2=4$$", "result": "=\\sec^{4}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7AtrsReczPdUqIdn5/Ev5mFtNfVJqszzpMwg6spHO8F8gJ/ZZA32ZInFBpDtxBfiKn3Jyj2VJgBojKQHjINpkkETYvnjqwhFu6aWmDOYpAVMO/U6qyRZonn1WPQ/21pz3cKXVLLQXMot2gtnZH7qY5zkbLioo7YVO0SRnDAO/7Uokt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "=2\\left(\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Expand $$\\left(\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)\\cdot\\:2:{\\quad}2\\sec^{4}\\left(θ\\right)+4\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)$$", "input": "\\left(\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)\\cdot\\:2", "result": "=2\\sec^{4}\\left(θ\\right)+4\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)", "steps": [ { "type": "step", "result": "=2\\left(\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the distributive law: $$a\\left(b+c\\right)=ab+ac$$", "secondary": [ "$$a=2,\\:b=\\sec^{4}\\left(θ\\right),\\:c=2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)$$" ], "result": "=2\\sec^{4}\\left(θ\\right)+2\\cdot\\:2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=2\\sec^{4}\\left(θ\\right)+4\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7cTiUO1fOFax2wXHZewjhxEUAdh/O2PLAxzMjSpKce9YL4tWjzbYCZhBh/Hc2rUXacMKakGXZAry+N/hvJOodjXCQoYlYQ8U+Tfyx0kyzI8iF3F2TkMs7mKjIsUPAR/Vd82dH/qvnCC7/oZy4HZlX2aHh6acbPp8G70jj/skoFx7vbBmbuQNTF0TphKZ8Ruvacu+LCkBvFQsw7xBL2Ys2TUYb1899n2cPVSJTSzzdZzp+38s+uQz1W6FLDk3KkIYWhJuRFkG6Ai089lDjeB1lMbCI2sSeA74029n2yo277ZU=" } }, { "type": "interim", "title": "Rewrite using trig identities", "input": "2\\sec^{4}\\left(θ\\right)+4\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)", "result": "=-4\\sec^{2}\\left(θ\\right)+6\\sec^{4}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Use the Pythagorean identity: $$\\tan^{2}\\left(x\\right)+1=\\sec^{2}\\left(x\\right)$$", "secondary": [ "$$\\tan^{2}\\left(x\\right)=\\sec^{2}\\left(x\\right)-1$$" ], "result": "=2\\sec^{4}\\left(θ\\right)+4\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right)" }, { "type": "interim", "title": "Simplify $$2\\sec^{4}\\left(θ\\right)+4\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right):{\\quad}6\\sec^{4}\\left(θ\\right)-4\\sec^{2}\\left(θ\\right)$$", "input": "2\\sec^{4}\\left(θ\\right)+4\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right)", "result": "=6\\sec^{4}\\left(θ\\right)-4\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "interim", "title": "Expand $$4\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right):{\\quad}4\\sec^{4}\\left(θ\\right)-4\\sec^{2}\\left(θ\\right)$$", "input": "4\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right)", "result": "=2\\sec^{4}\\left(θ\\right)+4\\sec^{4}\\left(θ\\right)-4\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b-c\\right)=ab-ac$$", "secondary": [ "$$a=4\\sec^{2}\\left(θ\\right),\\:b=\\sec^{2}\\left(θ\\right),\\:c=1$$" ], "result": "=4\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)-4\\sec^{2}\\left(θ\\right)\\cdot\\:1", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "result": "=4\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)-4\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$4\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)-4\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right):{\\quad}4\\sec^{4}\\left(θ\\right)-4\\sec^{2}\\left(θ\\right)$$", "input": "4\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)-4\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right)", "result": "=4\\sec^{4}\\left(θ\\right)-4\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "interim", "title": "$$4\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)=4\\sec^{4}\\left(θ\\right)$$", "input": "4\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)=\\:\\sec^{2+2}\\left(θ\\right)$$" ], "result": "=4\\sec^{2+2}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+2=4$$", "result": "=4\\sec^{4}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vjDr4p6TvcgiSLV1uDo4U9JYPDPDesonRI520G8zYRPehkKrn0era9rz8TlL+x/vMHVZf8xiryCPkbUIYXhiIHRfFgXB+vlRb/AHCJklpF9e7YF3TFAJSjcLX2VvhbubqX/cSuhK5Ty2xqd/IdNr6zlZ8CVoKZl2BAoDdAtcFNxU2fg2Eo8khoQfnK/P3ock" } }, { "type": "interim", "title": "$$4\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right)=4\\sec^{2}\\left(θ\\right)$$", "input": "4\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:1=4$$", "result": "=4\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7xF9rlQkuAOrRzKTf6eQs7fgkIqVBCzsTrV5i/ECNCw8tOtZYwUjyXhDTsNnn6Elr/BCsllMMEyc8UGKoUwuYxu6vfsn/ogOG+fHMfNQh9JeZfoRPsbCjih+xtMTOMGl4tzgLeG7CLgj8aHpc44iME/LWpEthEIQrAjNHY4qDKt/ET1Ehq4mhUZ9DWVx2az0M" } }, { "type": "step", "result": "=4\\sec^{4}\\left(θ\\right)-4\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vjDr4p6TvcgiSLV1uDo4U/jbMbEoNNbIR/7Q84OynCfTLx8mOdHYVzxX643JqKFIvTZkYD4gUeSoFrhGGhxq98kwYX4VRPb1geKkn+NU7mcE3SiHEhDlxIXJDVY+PPdvgQUxJPyUNnGfVirkcwpVO84fwxlvni/T8DIWnD0mw9U6FFV/0Hm6Y1hI7VaPzzKhkSoZP7CyArJlOo0FRny79Q==" } }, { "type": "step", "primary": "Add similar elements: $$2\\sec^{4}\\left(θ\\right)+4\\sec^{4}\\left(θ\\right)=6\\sec^{4}\\left(θ\\right)$$", "result": "=6\\sec^{4}\\left(θ\\right)-4\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7f48emNsIl4MI4hQZuZje4penFTHzh6X+DX3uvPdqYy7umM/GfBKdAtyuCOAeIKZmAJYpRu9XpYrd8NSAW2DdD0c/CXknyRKFbCevkP3vkuVyZXAw9UlCE0mEZtdsFPrV7q9+yf+iA4b58cx81CH0l/C30sSftAIFS6Qkpy19Ikp0x8irWzKvvppKMUvqTrnrUObZozYPTEmgGDR5V3grHDoUVX/QebpjWEjtVo/PMqGRKhk/sLICsmU6jQVGfLv1" } } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7xqDNX42OyJjPevbF+7IBng7708BkbVE1gb8/0kMEgzvXUIQv0aG2ncbDgSWX/DWycNZx/2EDub2dSsLw3NGg9xLC2T8NFZvTA+pff5AdXPdX9lRN4mlYq/zI0dgxEXGdPvdnhi0s3lZ+DJ7rEiTeqenxKTmAB1ahPFE542Dd/TeutOLi0kB5xHysaBsHIgntIaTy8gmKy5PY3el+RbUmlkmFP65WPUHCIGTpzT/EAccetg60+ZUlqP5jB+ReldUA5AMRxXWt+t9va0qZyagxyQ==" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7xI7dknI212svHgZHHIcoRvkdSragnZ5K8+WMRM5qjJ6ETkUamdUz59t0aHpV3uN0rE0Ib0rKQ2oM97vl/gDbowCWKUbvV6WK3fDUgFtg3Q+MfZ1BFRhhGnQf9q28g9nK/SzNF20Xrftd883pI0RALQ5+VkRmueXOP55Piqx5MyWBBTEk/JQ2cZ9WKuRzClU7m4RppR4RRNJR6+bqv/lHExUnakC9Y1jhr4uI97E9yvalDk6K3L00/zgNSSliALTtnpy3tAot7+5PIP9fBKZw2Llxun299aFzbjCyRFblUKE=" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{d^{5}}{dθ^{5}}\\left(-4\\sec^{2}\\left(θ\\right)+6\\sec^{4}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(-4\\sec^{2}\\left(θ\\right)+6\\sec^{4}\\left(θ\\right)\\right)=-8\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+24\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(-4\\sec^{2}\\left(θ\\right)+6\\sec^{4}\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=-\\frac{d}{dθ}\\left(4\\sec^{2}\\left(θ\\right)\\right)+\\frac{d}{dθ}\\left(6\\sec^{4}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(4\\sec^{2}\\left(θ\\right)\\right)=8\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(4\\sec^{2}\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=4\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Apply the chain rule:$${\\quad}2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{2},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{2}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{2}\\right)=2u$$", "input": "\\frac{d}{du}\\left(u^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2u^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2u", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYr+VZUwBnLdzbS6DZQ+f4s+k3hxk9aCfAWodBRxXgUexMchyqTAJWrzJaDbnNcFsJUeCBKuYKgaNJ253gLI69U79qbCA2QqVmvm3jGRXZ2ppvbGT4j1utMEkCDH25m/vlQ==" } }, { "type": "step", "result": "=2u\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgCSXIThcWH58oYoMPt18lzFqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidZClirlWNFPnHR1rvdyoFLI7IC+oqnwb1T6nPLpiw1eysw2NXPd8jZnr3fcPpqc7ejeh7+jKEzLb7VNCEMF3Z/Maswa5oIagKQFOfrnKm5+Gbb/PxX3awut5KxRxyZf8A=" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=4\\cdot\\:2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$4\\cdot\\:2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}8\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "4\\cdot\\:2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=8\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:2=8$$", "result": "=8\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{1+1}\\left(θ\\right)$$" ], "result": "=8\\sec^{1+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=8\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7NqBYpvGKZC5lMRQos2qph5K/zREn/U8meKQ3TvRjTQz8cZMsH3IX6lxp+ANHGbuCq47vuWedXv2WUg94ER8IwTfgzf+5OuhkULh+nunjrqae0Rv083m3K3ie3DVArOu08LfSxJ+0AgVLpCSnLX0iSqXiJoCK22T5YMsRbcSz8DsyFZaieHAvmtaw5r6RKguDHxgfR9uWXPlGtc+ysPJIG+IASZeFjDtawNGt9P21GJY=" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(6\\sec^{4}\\left(θ\\right)\\right)=24\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(6\\sec^{4}\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=6\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Apply the chain rule:$${\\quad}4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{4},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{4}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{4}\\right)=4u^{3}$$", "input": "\\frac{d}{du}\\left(u^{4}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=4u^{4-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=4u^{3}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYl3oEe4EC6n4B8Xgz0cKnkqk3hxk9aCfAWodBRxXgUexuzo0Fl6VhneutEteidn+Kv8//6/nV5O4fb8Xgwi7mapNFxUvwBeni+JEIFAdbegzYEOcQgyW4n51cR7AeMAm7Q==" } }, { "type": "step", "result": "=4u^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgA2CO5cm49xNEqFV0DMaZJRqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidVduWEnoDYDg06IbaTsp9TFtqiuNhFTvmF8ZCDqWZNoubfnOah8gAvLMKOeNMctC5tVYZ/3r82LxPimKUFXjQFStvD8GKDNJJN8qiu0XRcVZAYAvbRKqkRVDaDBHkSBtDwYHkfmHHTze3JXk3LjiJp" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=6\\cdot\\:4\\left(\\sec\\left(θ\\right)\\right)^{3}\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$6\\cdot\\:4\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}24\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "6\\cdot\\:4\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=24\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$6\\cdot\\:4=24$$", "result": "=24\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{3+1}\\left(θ\\right)$$" ], "result": "=24\\sec^{3+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$3+1=4$$", "result": "=24\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l3FkS70/4BFsebxcaZbHnTNSoP5XQt+n/5y371e8QHJROlTYN926lBwmiJdD5HUSzRqDxPUzBN6vjj5oJL9kUIdu2MPseykkfgn/Trt10KGp9ej8WEAztP3UNEC9n7jjZEt3ZXAiqUE0HIXrrrezJHZKl2mV4AlztiApTAFigz07AW6Jz0sXkPNp6b6xpHMqM1Kg/ldC36f/nLfvV7xActytFlzCV8oq55bHLwXbb0c=" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=-8\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+24\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{d^{4}}{dθ^{4}}\\left(-8\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+24\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(-8\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+24\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)=-8\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)+24\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(-8\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+24\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=-\\frac{d}{dθ}\\left(8\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+\\frac{d}{dθ}\\left(24\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(8\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\right)=8\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(8\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=8\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the Product Rule: $$\\left(f{\\cdot}g\\right)'=f'{\\cdot}g+f{\\cdot}g'$$", "secondary": [ "$$f=\\sec^{2}\\left(θ\\right),\\:g=\\tan\\left(θ\\right)$$" ], "result": "=8\\left(\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)\\tan\\left(θ\\right)+\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)\\sec^{2}\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Product%20Rule", "practiceTopic": "Product Rule" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{2},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{2}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{2}\\right)=2u$$", "input": "\\frac{d}{du}\\left(u^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2u^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2u", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYr+VZUwBnLdzbS6DZQ+f4s+k3hxk9aCfAWodBRxXgUexMchyqTAJWrzJaDbnNcFsJUeCBKuYKgaNJ253gLI69U79qbCA2QqVmvm3jGRXZ2ppvbGT4j1utMEkCDH25m/vlQ==" } }, { "type": "step", "result": "=2u\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgCSXIThcWH58oYoMPt18lzFqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidZClirlWNFPnHR1rvdyoFLI7IC+oqnwb1T6nPLpiw1eysw2NXPd8jZnr3fcPpqc7ejeh7+jKEzLb7VNCEMF3Z/Maswa5oIagKQFOfrnKm5+Gbb/PxX3awut5KxRxyZf8A=" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{1+1}\\left(θ\\right)$$" ], "result": "=2\\sec^{1+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7aY4rYekMZhUfP6sZ6mfqd6Lsz0lt1XLEr88Kc/cjJo5V00rpv8+ZC6TM10tVCSHspqrCcTwb/I0f8OwMjuU6eBAUa+AHsHd5OGpMiuFITjOuqvW3hAHH635V/vEEYIatHimBRYRqHSWeJkuUPhfTC5LQ0B1JNgsb5SXzbS+SCnAfGB9H25Zc+Ua1z7Kw8kgb4gBJl4WMO1rA0a30/bUYlg==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "result": "=\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYs0Vsx3c0nt53Utu5DgHTe3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK00HV/bLwwglXz/2YNx2i4JHggSrmCoGjSdud4CyOvVOgdQN0k+N3NHNAkEIoLrhB631k80nn+ODwI4ID0kCF5+z7zRz81SqRD02HEUE4Wqh" } }, { "type": "step", "result": "=8\\left(2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Simplify $$8\\left(2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\right):{\\quad}8\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)$$", "input": "8\\left(2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\right)", "result": "=8\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "$$2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)=2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)$$", "input": "2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\tan\\left(θ\\right)\\tan\\left(θ\\right)=\\:\\tan^{1+1}\\left(θ\\right)$$" ], "result": "=2\\sec^{2}\\left(θ\\right)\\tan^{1+1}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Md1SJKHYLxhOYRRbsLTu+axUZgt6FTmFSHnRsotkqemLeJKmeakO1+Q3aSIldG41s7BwSVdKEYSqn+4fKAQoGk0xmFIl1KTO5aAkhaHvKzBALhShbMGZIZsZWTO721HsWI4dG63F/aSz/PFDvu5FnO3sngYNzcydIedGQjjtM8sBMUhkLMqDD6KHvgZAgKMdRQzIIgXxnWeDhDTk2Lfum8RPUSGriaFRn0NZXHZrPQw=" } }, { "type": "interim", "title": "$$\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)=\\sec^{4}\\left(θ\\right)$$", "input": "\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)=\\:\\sec^{2+2}\\left(θ\\right)$$" ], "result": "=\\sec^{2+2}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+2=4$$", "result": "=\\sec^{4}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7AtrsReczPdUqIdn5/Ev5mFtNfVJqszzpMwg6spHO8F8gJ/ZZA32ZInFBpDtxBfiKn3Jyj2VJgBojKQHjINpkkETYvnjqwhFu6aWmDOYpAVMO/U6qyRZonn1WPQ/21pz3cKXVLLQXMot2gtnZH7qY5zkbLioo7YVO0SRnDAO/7Uokt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "=8\\left(\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Simplify $$2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{4}\\left(θ\\right):{\\quad}3\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)$$", "input": "2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{4}\\left(θ\\right)", "result": "=8\\left(3\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Rewrite using trig identities", "input": "\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)", "result": "=-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Use the Pythagorean identity: $$\\tan^{2}\\left(x\\right)+1=\\sec^{2}\\left(x\\right)$$", "secondary": [ "$$\\tan^{2}\\left(x\\right)=\\sec^{2}\\left(x\\right)-1$$" ], "result": "=\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right)" }, { "type": "interim", "title": "Simplify $$\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right):{\\quad}3\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)$$", "input": "\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right)", "result": "=3\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "interim", "title": "Expand $$2\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right):{\\quad}2\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)$$", "input": "2\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right)", "result": "=\\sec^{4}\\left(θ\\right)+2\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b-c\\right)=ab-ac$$", "secondary": [ "$$a=2\\sec^{2}\\left(θ\\right),\\:b=\\sec^{2}\\left(θ\\right),\\:c=1$$" ], "result": "=2\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)\\cdot\\:1", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "result": "=2\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)-2\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$2\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)-2\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right):{\\quad}2\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)$$", "input": "2\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)-2\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right)", "result": "=2\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "interim", "title": "$$2\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)=2\\sec^{4}\\left(θ\\right)$$", "input": "2\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)=\\:\\sec^{2+2}\\left(θ\\right)$$" ], "result": "=2\\sec^{2+2}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+2=4$$", "result": "=2\\sec^{4}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Md1SJKHYLxhOYRRbsLTu+dJYPDPDesonRI520G8zYRPehkKrn0era9rz8TlL+x/vyvn14PieOyuPP98CkiPIaXRfFgXB+vlRb/AHCJklpF+qQC22PXYzQL8KzAhJCzszqX/cSuhK5Ty2xqd/IdNr67A2LXxYB8f3C3KpWaLgll1U2fg2Eo8khoQfnK/P3ock" } }, { "type": "interim", "title": "$$2\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right)=2\\sec^{2}\\left(θ\\right)$$", "input": "2\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=2\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ht1jgkupea8+jUgT+M/VFPgkIqVBCzsTrV5i/ECNCw8tOtZYwUjyXhDTsNnn6Elr8U8lPA4O6WXezAN1J2P7Ue6vfsn/ogOG+fHMfNQh9Je7opJQAzFYPpqn/2bLavXutzgLeG7CLgj8aHpc44iMEx91+u64Nzpp2/441mWGcQ/ET1Ehq4mhUZ9DWVx2az0M" } }, { "type": "step", "result": "=2\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Md1SJKHYLxhOYRRbsLTu+fjbMbEoNNbIR/7Q84OynCfTLx8mOdHYVzxX643JqKFI1Jetp9AvoQcY2QSDwFJhz1kUvGSawlk7umQ8kurx5doE3SiHEhDlxIXJDVY+PPdvgQUxJPyUNnGfVirkcwpVO6RMkE3rByM/Y1Dxxfs3xFw6FFV/0Hm6Y1hI7VaPzzKhkSoZP7CyArJlOo0FRny79Q==" } }, { "type": "step", "primary": "Add similar elements: $$\\sec^{4}\\left(θ\\right)+2\\sec^{4}\\left(θ\\right)=3\\sec^{4}\\left(θ\\right)$$", "result": "=3\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ARkhlGZdazrdxNLzhlqePtRiXXTzenxbt00ahL9sDS5L0MjjUe01mdeSwRMoYPSBLTrWWMFI8l4Q07DZ5+hJazaxvejhvTJvqoS105+T9+xc3qH/hVuCay9igw/1IwZBNySLUGdPDJeW3uY1XSs479bA+zX4bD3u3gx65o2NJhP8KqIsxz0lwScBMkZS31XhRCRwG0yFaMyhS3uC5/uUeMvqafR00cq/gMXQBnztqPn0yTzBsNnlyCSr/zrcxdQG" } } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7FkSOkw7Nc+ISjlBZ7XiESO/dTLaj0YrNSw6+/ogyGX8k+jBZ4XxujMMeS3Qvsi6m7aEe0hvT9VtxXei653t2zQi9Vnn4imguIoWn3N1eqjdGIkb3QW2w7y4N83RSktbc/kUupyp7xXxB8YmKEQ5HoPs4SAUa1zTfp2gkDWnxoYVtNtNgAtA4xHxju0w39zFkZEt3ZXAiqUE0HIXrrrezJJi0cdkgnw2HPjRVpcHMxZbIciYVzjaAolEaNkVONtBBRppTNgMFons9bMkTzIHooA==" } } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7JzuoJyZ0ket+0by442o6nfkdSragnZ5K8+WMRM5qjJ6ETkUamdUz59t0aHpV3uN0rE0Ib0rKQ2oM97vl/gDbowCWKUbvV6WK3fDUgFtg3Q+nr1JqgQ+t+tehVxpQusgo+5k7zGGz9wmA2lyQsVWy+j6F2h9EjPuS9/S594y1Nd1FKk3fejFkyiOiq9iG9IkAIuSmJUDtLxRPyJ57Nkq7qjUyCLMvQZ782OaKpAVh7FOsVGYLehU5hUh50bKLZKnpo1d4a+pBCGznOTLxP2sni3enTnyZLeuZs6pBwuQMplUkt3WiGR7ZaCaXvz77bMjS" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(24\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)=24\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(24\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=24\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the Product Rule: $$\\left(f{\\cdot}g\\right)'=f'{\\cdot}g+f{\\cdot}g'$$", "secondary": [ "$$f=\\sec^{4}\\left(θ\\right),\\:g=\\tan\\left(θ\\right)$$" ], "result": "=24\\left(\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)\\tan\\left(θ\\right)+\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)\\sec^{4}\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Product%20Rule", "practiceTopic": "Product Rule" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)=4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{4},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{4}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{4}\\right)=4u^{3}$$", "input": "\\frac{d}{du}\\left(u^{4}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=4u^{4-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=4u^{3}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYl3oEe4EC6n4B8Xgz0cKnkqk3hxk9aCfAWodBRxXgUexuzo0Fl6VhneutEteidn+Kv8//6/nV5O4fb8Xgwi7mapNFxUvwBeni+JEIFAdbegzYEOcQgyW4n51cR7AeMAm7Q==" } }, { "type": "step", "result": "=4u^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgA2CO5cm49xNEqFV0DMaZJRqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidVduWEnoDYDg06IbaTsp9TFtqiuNhFTvmF8ZCDqWZNoubfnOah8gAvLMKOeNMctC5tVYZ/3r82LxPimKUFXjQFStvD8GKDNJJN8qiu0XRcVZAYAvbRKqkRVDaDBHkSBtDwYHkfmHHTze3JXk3LjiJp" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$4\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "4\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{3+1}\\left(θ\\right)$$" ], "result": "=4\\sec^{3+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$3+1=4$$", "result": "=4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7p9w76+/nvHitDrMIupHrrrPmAPxuc/RFHWXkB8A+mnqLeJKmeakO1+Q3aSIldG41PCyA7ixjCdfDYe+yac7KOvOIrkYdKW3Q58EnXaMU+7oslHPSvPf5UfEs5T+IFZ4XHjb2+5NLFZrsH9fcPWg/TXL71sHPAsAQtYZvjpaGSRwboK/Igc1IRC3EGruCoRzV1Zrc/SYRIR8JuxYpD+PJ+Q==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "result": "=\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYs0Vsx3c0nt53Utu5DgHTe3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK00HV/bLwwglXz/2YNx2i4JHggSrmCoGjSdud4CyOvVOgdQN0k+N3NHNAkEIoLrhB631k80nn+ODwI4ID0kCF5+z7zRz81SqRD02HEUE4Wqh" } }, { "type": "step", "result": "=24\\left(4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Simplify $$24\\left(4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right):{\\quad}24\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)$$", "input": "24\\left(4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right)", "result": "=24\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "$$4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)=4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)$$", "input": "4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\tan\\left(θ\\right)\\tan\\left(θ\\right)=\\:\\tan^{1+1}\\left(θ\\right)$$" ], "result": "=4\\sec^{4}\\left(θ\\right)\\tan^{1+1}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7GHerUg4zB8o0EeBO6BKV16xUZgt6FTmFSHnRsotkqemLeJKmeakO1+Q3aSIldG41PCyA7ixjCdfDYe+yac7KOvOIrkYdKW3Q58EnXaMU+7pALhShbMGZIZsZWTO721HsXaBT4KNKuhZZSFsFWnKRStdNJOkRdCtrWB4iMQVmMwBoAWE3NiWlLPrDNnXLOZLK3tKI9zGyi4xCTaccmboyHcRPUSGriaFRn0NZXHZrPQw=" } }, { "type": "interim", "title": "$$\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)=\\sec^{6}\\left(θ\\right)$$", "input": "\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)=\\:\\sec^{2+4}\\left(θ\\right)$$" ], "result": "=\\sec^{2+4}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+4=6$$", "result": "=\\sec^{6}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7AtrsReczPdUqIdn5/Ev5mLBxohAhN47qWD2nlwOHqfkgJ/ZZA32ZInFBpDtxBfiKn3Jyj2VJgBojKQHjINpkkG6JHhxn2McSW7OxLgQBi64O/U6qyRZonn1WPQ/21pz3xMe3sR8ZMoA2Ynlp2W853t3OcQiSW6rZ4zI81UhADI4kt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "=24\\left(\\sec^{6}\\left(θ\\right)+4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7jhDHfABKmSqcJJ1fAFAsGBiMJmjxigW8LortZKvrqEbWPF6oWuXggupsFvXrooIqxMe3sR8ZMoA2Ynlp2W853tMvHyY50dhXPFfrjcmooUg3CXpyb+Iu3cMrCn545PU8uO1NenZJP60hEwBH05kmlvOIrkYdKW3Q58EnXaMU+7r4ZAjrJM4mwvgfYovsxoopgQUxJPyUNnGfVirkcwpVOxKnUVO0TgjLT50Mmk0DzLYh0PvFonSnuZi7Z6ItIhmYhYbzXwn7iok10o2w+NJLRiZIb9CFuBJdPyi+MBoLCMd/LnYygZMYhjzpUCXiHwCy" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=-8\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)+24\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{d^{3}}{dθ^{3}}\\left(-8\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)+24\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(-8\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)+24\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)\\right)=-8\\left(-4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+24\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(-8\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)+24\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=-\\frac{d}{dθ}\\left(8\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)\\right)+\\frac{d}{dθ}\\left(24\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(8\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)\\right)=8\\left(-4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(8\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=8\\frac{d}{dθ}\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=8\\left(-\\frac{d}{dθ}\\left(2\\sec^{2}\\left(θ\\right)\\right)+\\frac{d}{dθ}\\left(3\\sec^{4}\\left(θ\\right)\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(2\\sec^{2}\\left(θ\\right)\\right)=4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(2\\sec^{2}\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=2\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Apply the chain rule:$${\\quad}2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{2},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{2}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{2}\\right)=2u$$", "input": "\\frac{d}{du}\\left(u^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2u^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2u", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYr+VZUwBnLdzbS6DZQ+f4s+k3hxk9aCfAWodBRxXgUexMchyqTAJWrzJaDbnNcFsJUeCBKuYKgaNJ253gLI69U79qbCA2QqVmvm3jGRXZ2ppvbGT4j1utMEkCDH25m/vlQ==" } }, { "type": "step", "result": "=2u\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgCSXIThcWH58oYoMPt18lzFqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidZClirlWNFPnHR1rvdyoFLI7IC+oqnwb1T6nPLpiw1eysw2NXPd8jZnr3fcPpqc7ejeh7+jKEzLb7VNCEMF3Z/Maswa5oIagKQFOfrnKm5+Gbb/PxX3awut5KxRxyZf8A=" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=2\\cdot\\:2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$2\\cdot\\:2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "2\\cdot\\:2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=4\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{1+1}\\left(θ\\right)$$" ], "result": "=4\\sec^{1+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s71jEHNygbGF2qQyJrTxQFvpK/zREn/U8meKQ3TvRjTQz8cZMsH3IX6lxp+ANHGbuCq47vuWedXv2WUg94ER8IwfulLd9uW7xFGDkqRxB3vSSe0Rv083m3K3ie3DVArOu08LfSxJ+0AgVLpCSnLX0iSnTHyKtbMq++mkoxS+pOuesyFZaieHAvmtaw5r6RKguDHxgfR9uWXPlGtc+ysPJIG+IASZeFjDtawNGt9P21GJY=" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(3\\sec^{4}\\left(θ\\right)\\right)=12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(3\\sec^{4}\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=3\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Apply the chain rule:$${\\quad}4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{4},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{4}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{4}\\right)=4u^{3}$$", "input": "\\frac{d}{du}\\left(u^{4}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=4u^{4-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=4u^{3}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYl3oEe4EC6n4B8Xgz0cKnkqk3hxk9aCfAWodBRxXgUexuzo0Fl6VhneutEteidn+Kv8//6/nV5O4fb8Xgwi7mapNFxUvwBeni+JEIFAdbegzYEOcQgyW4n51cR7AeMAm7Q==" } }, { "type": "step", "result": "=4u^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgA2CO5cm49xNEqFV0DMaZJRqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidVduWEnoDYDg06IbaTsp9TFtqiuNhFTvmF8ZCDqWZNoubfnOah8gAvLMKOeNMctC5tVYZ/3r82LxPimKUFXjQFStvD8GKDNJJN8qiu0XRcVZAYAvbRKqkRVDaDBHkSBtDwYHkfmHHTze3JXk3LjiJp" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=3\\cdot\\:4\\left(\\sec\\left(θ\\right)\\right)^{3}\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$3\\cdot\\:4\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "3\\cdot\\:4\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$3\\cdot\\:4=12$$", "result": "=12\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{3+1}\\left(θ\\right)$$" ], "result": "=12\\sec^{3+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$3+1=4$$", "result": "=12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7jFGb0q9vd+kdWvVYeHJHbDNSoP5XQt+n/5y371e8QHJROlTYN926lBwmiJdD5HUSzRqDxPUzBN6vjj5oJL9kUFPzsE52Mw7P00gxEE61byKp9ej8WEAztP3UNEC9n7jjZEt3ZXAiqUE0HIXrrrezJHZKl2mV4AlztiApTAFigz2gNkWvAT/c3O3uDO/qxcKwM1Kg/ldC36f/nLfvV7xActytFlzCV8oq55bHLwXbb0c=" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=8\\left(-4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(24\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)\\right)=24\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(24\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=24\\frac{d}{dθ}\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=24\\left(\\frac{d}{dθ}\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)+\\frac{d}{dθ}\\left(\\sec^{6}\\left(θ\\right)\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)=4\\left(4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=4\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the Product Rule: $$\\left(f{\\cdot}g\\right)'=f'{\\cdot}g+f{\\cdot}g'$$", "secondary": [ "$$f=\\sec^{4}\\left(θ\\right),\\:g=\\tan^{2}\\left(θ\\right)$$" ], "result": "=4\\left(\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)\\tan^{2}\\left(θ\\right)+\\frac{d}{dθ}\\left(\\tan^{2}\\left(θ\\right)\\right)\\sec^{4}\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Product%20Rule", "practiceTopic": "Product Rule" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)=4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{4},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{4}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{4}\\right)=4u^{3}$$", "input": "\\frac{d}{du}\\left(u^{4}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=4u^{4-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=4u^{3}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYl3oEe4EC6n4B8Xgz0cKnkqk3hxk9aCfAWodBRxXgUexuzo0Fl6VhneutEteidn+Kv8//6/nV5O4fb8Xgwi7mapNFxUvwBeni+JEIFAdbegzYEOcQgyW4n51cR7AeMAm7Q==" } }, { "type": "step", "result": "=4u^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgA2CO5cm49xNEqFV0DMaZJRqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidVduWEnoDYDg06IbaTsp9TFtqiuNhFTvmF8ZCDqWZNoubfnOah8gAvLMKOeNMctC5tVYZ/3r82LxPimKUFXjQFStvD8GKDNJJN8qiu0XRcVZAYAvbRKqkRVDaDBHkSBtDwYHkfmHHTze3JXk3LjiJp" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$4\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "4\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{3+1}\\left(θ\\right)$$" ], "result": "=4\\sec^{3+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$3+1=4$$", "result": "=4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7p9w76+/nvHitDrMIupHrrrPmAPxuc/RFHWXkB8A+mnqLeJKmeakO1+Q3aSIldG41PCyA7ixjCdfDYe+yac7KOvOIrkYdKW3Q58EnXaMU+7oslHPSvPf5UfEs5T+IFZ4XHjb2+5NLFZrsH9fcPWg/TXL71sHPAsAQtYZvjpaGSRwboK/Igc1IRC3EGruCoRzV1Zrc/SYRIR8JuxYpD+PJ+Q==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan^{2}\\left(θ\\right)\\right)=2\\tan\\left(θ\\right)\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan^{2}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}2\\tan\\left(θ\\right)\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan^{2}\\left(θ\\right)\\right)", "result": "=2\\tan\\left(θ\\right)\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{2},\\:\\:u=\\tan\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{2}\\right)\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{2}\\right)=2u$$", "input": "\\frac{d}{du}\\left(u^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2u^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2u", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYr+VZUwBnLdzbS6DZQ+f4s+k3hxk9aCfAWodBRxXgUexMchyqTAJWrzJaDbnNcFsJUeCBKuYKgaNJ253gLI69U79qbCA2QqVmvm3jGRXZ2ppvbGT4j1utMEkCDH25m/vlQ==" } }, { "type": "step", "result": "=2u\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\tan\\left(θ\\right)$$", "result": "=2\\tan\\left(θ\\right)\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYux7TmDJ3fopnHGBxoqO+paSXIThcWH58oYoMPt18lzFqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidIn+HfeW3QQwAkRa15J0NSAlj501WLIv46IdbBaParYow2V0WdtbiweShXv1KOwqCjeh7+jKEzLb7VNCEMF3Z/Maswa5oIagKQFOfrnKm5+Gbb/PxX3awut5KxRxyZf8A=" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "result": "=\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYs0Vsx3c0nt53Utu5DgHTe3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK00HV/bLwwglXz/2YNx2i4JHggSrmCoGjSdud4CyOvVOgdQN0k+N3NHNAkEIoLrhB631k80nn+ODwI4ID0kCF5+z7zRz81SqRD02HEUE4Wqh" } }, { "type": "step", "result": "=2\\tan\\left(θ\\right)\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=4\\left(4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{2}\\left(θ\\right)+2\\tan\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Simplify $$4\\left(4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{2}\\left(θ\\right)+2\\tan\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right):{\\quad}4\\left(4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)$$", "input": "4\\left(4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{2}\\left(θ\\right)+2\\tan\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right)", "result": "=4\\left(4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "$$4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{2}\\left(θ\\right)=4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)$$", "input": "4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\tan\\left(θ\\right)\\tan^{2}\\left(θ\\right)=\\:\\tan^{1+2}\\left(θ\\right)$$" ], "result": "=4\\sec^{4}\\left(θ\\right)\\tan^{1+2}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+2=3$$", "result": "=4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7GHerUg4zB8o0EeBO6BKV162cmBc6Js8szDUhC2Hxdqi5HNbiQL8+p2P+VJTsJbU7q47vuWedXv2WUg94ER8IwVADhNIEz39BpXewdhMCBb+bbDkw/o6UIkLjzolgTAsqkuHx4ulZqfxZHIsiQUpV1nyo9elyUFFpcPZW02cAt7EtbJdlgVP3iSyKFg6v1zm6cUwY1HWd9EJWOCb4O+kauMdM6yd/xAZrmYAkuRY8/rYkt3WiGR7ZaCaXvz77bMjS" } }, { "type": "interim", "title": "$$2\\tan\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)=2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "2\\tan\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)=\\:\\sec^{2+4}\\left(θ\\right)$$" ], "result": "=2\\tan\\left(θ\\right)\\sec^{2+4}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+4=6$$", "result": "=2\\tan\\left(θ\\right)\\sec^{6}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7OyotidIxGYYp2AND0rbhNaCGttAzkBWvgG18MKDLv3Rdbcltti6PzkTDSwkmBTY6q47vuWedXv2WUg94ER8IwSIYCdSNn6m2gNv8FkjgS1XBL9WNCz07kx2T4iGijNtQjT/+cTnIKvMqhzKaFBbxvzSei4QZxxrazp/cx7cYZvbVeYdTw2sAeZ9o0nz1uUihamteA8UyY/eLm9jagP57hkwLhsnE4D1C6IGVVz/pqlY=" } }, { "type": "step", "result": "=4\\left(2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)+4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7iza9NvP7nq2MIMeLHhROsvkdSragnZ5K8+WMRM5qjJ4/XhyePE1SthuGDJcaFMnncKXVLLQXMot2gtnZH7qY51d1u0XlM+NdfDNBhSESuvlV00rpv8+ZC6TM10tVCSHsi/RlR31DZv0142/jAcann2ASrhThz5UQhS79xvpIR8tIoCq5NivR6sCY2Sfa33kc8jzDW/n9HorFBpgomiqIe+5byrQDQVCXUD0vH/fvOdxyhd7tjiG+GxQNxDvGkZUlqqyy15Nuz77zqSegKeIT2iKI9Km/ZvzCnLF9eqmklKHqRJDJ/DeTzEayKJpmRlyWQi730E1b9JEo9956Qy3sPLLWww1IW5GUOlAGrcUL2OqwiNrEngO+NNvZ9sqNu+2V" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec^{6}\\left(θ\\right)\\right)=6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{6}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}6\\left(\\sec\\left(θ\\right)\\right)^{5}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{6}\\left(θ\\right)\\right)", "result": "=6\\left(\\sec\\left(θ\\right)\\right)^{5}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{6},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{6}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{6}\\right)=6u^{5}$$", "input": "\\frac{d}{du}\\left(u^{6}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=6u^{6-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=6u^{5}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYgOK76JOSNqGa1B1wU6ozWWk3hxk9aCfAWodBRxXgUexqSoDkuZ4Ab1St3bM8oqKcP8//6/nV5O4fb8Xgwi7mapNFxUvwBeni+JEIFAdbegzHfwjUTnSu4cWqw+8m79+NA==" } }, { "type": "step", "result": "=6u^{5}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=6\\left(\\sec\\left(θ\\right)\\right)^{5}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgBaJTGF5uPwSMzxoSyJNVZsqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKieMSe35zYqdvyntnEYxaQ88jJq7JZFtKgvxzzzuaqTGrebfnOah8gAvLMKOeNMctC5tVYZ/3r82LxPimKUFXjQFStvD8GKDNJJN8qiu0XRcVZAYAvbRKqkRVDaDBHkSBtDwYHkfmHHTze3JXk3LjiJp" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=6\\left(\\sec\\left(θ\\right)\\right)^{5}\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$6\\sec^{5}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "6\\sec^{5}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{5}\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{5+1}\\left(θ\\right)$$" ], "result": "=6\\sec^{5+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$5+1=6$$", "result": "=6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7QMJmgA2Sx2hYpGS5j/h1NLPmAPxuc/RFHWXkB8A+mnqLeJKmeakO1+Q3aSIldG417JiujyaxykLR7Tv0Nr+wOGprXgPFMmP3i5vY2oD+e4YslHPSvPf5UfEs5T+IFZ4XHjb2+5NLFZrsH9fcPWg/TRiIlnZT/wCWlhB0uSVNtvx/uR/EZqc6EbI3tnxSI5001Zrc/SYRIR8JuxYpD+PJ+Q==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=24\\left(4\\left(4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Expand $$4\\left(4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "4\\left(4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=24\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Expand $$4\\left(4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right):{\\quad}16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+8\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "4\\left(4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)", "result": "=16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+8\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)+6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b+c\\right)=ab+ac$$", "secondary": [ "$$a=4,\\:b=4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right),\\:c=2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)$$" ], "result": "=4\\cdot\\:4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+4\\cdot\\:2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "interim", "title": "Simplify $$4\\cdot\\:4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+4\\cdot\\:2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+8\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "4\\cdot\\:4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+4\\cdot\\:2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+8\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:4=16$$", "result": "=16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+4\\cdot\\:2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:2=8$$", "result": "=16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+8\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7iza9NvP7nq2MIMeLHhROsoL8UIMToMV6hDKJhY+Nk7toxrHhSJArgBpsnuAV9UCfBgOJ+lz5D5t+hsCNYCoDMHWD310L1+P2yDQQfMEhENFHx6CvpVa90MvMzklpzJWXZAzoThz1g+TITtf+WJ7KiyRRh7kCo0obc/CTaNyPGkYuuAKnMbBNnPJ2x/QM7ydV1sD7NfhsPe7eDHrmjY0mEyzoLVQPyp9R+5GC0prpwlf2K8DiYaJsjNsjOcdOcdQLW6BryIKCtddVLZsZyZ1dV9DmtpfPTl/o4ZJKfQIdsQqwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "primary": "Add similar elements: $$8\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)+6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)=14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7iza9NvP7nq2MIMeLHhROsoL8UIMToMV6hDKJhY+Nk7toxrHhSJArgBpsnuAV9UCf8SVGtBi1ovwgT5vDB2QfkF3B8WOxYrIgIdHvusvufLAAlilG71elit3w1IBbYN0PumuR+HimzGDTYbsfY1Q6JXyo9elyUFFpcPZW02cAt7EXYiLGBL7PG2xAqJd1w6BXKK1Q1tVX7SUdYWONKsZTX66q9beEAcfrflX+8QRghq2CcvL2jSzw1VlRcGX4ka6qwZ5KNwm43WeyBMBBKL30QvI8w1v5/R6KxQaYKJoqiHsLiDrnJdtlx1I2mhkkUTElN8b/k79QNO9t6sAoPBR3zJYojigZ8i1ymB3KwWBswnjiAEmXhYw7WsDRrfT9tRiW" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=-8\\left(-4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+24\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{d^{2}}{dθ^{2}}\\left(-8\\left(-4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+24\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(-8\\left(-4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+24\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)\\right)=-8\\left(-4\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)+12\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)\\right)+24\\left(64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+132\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+14\\sec^{8}\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(-8\\left(-4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+24\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=-\\frac{d}{dθ}\\left(8\\left(-4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)\\right)+\\frac{d}{dθ}\\left(24\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(8\\left(-4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)\\right)=8\\left(-4\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)+12\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(8\\left(-4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=8\\frac{d}{dθ}\\left(-4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=8\\left(-\\frac{d}{dθ}\\left(4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+\\frac{d}{dθ}\\left(12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\right)=4\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=4\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the Product Rule: $$\\left(f{\\cdot}g\\right)'=f'{\\cdot}g+f{\\cdot}g'$$", "secondary": [ "$$f=\\sec^{2}\\left(θ\\right),\\:g=\\tan\\left(θ\\right)$$" ], "result": "=4\\left(\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)\\tan\\left(θ\\right)+\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)\\sec^{2}\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Product%20Rule", "practiceTopic": "Product Rule" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{2},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{2}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{2}\\right)=2u$$", "input": "\\frac{d}{du}\\left(u^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2u^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2u", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYr+VZUwBnLdzbS6DZQ+f4s+k3hxk9aCfAWodBRxXgUexMchyqTAJWrzJaDbnNcFsJUeCBKuYKgaNJ253gLI69U79qbCA2QqVmvm3jGRXZ2ppvbGT4j1utMEkCDH25m/vlQ==" } }, { "type": "step", "result": "=2u\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgCSXIThcWH58oYoMPt18lzFqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidZClirlWNFPnHR1rvdyoFLI7IC+oqnwb1T6nPLpiw1eysw2NXPd8jZnr3fcPpqc7ejeh7+jKEzLb7VNCEMF3Z/Maswa5oIagKQFOfrnKm5+Gbb/PxX3awut5KxRxyZf8A=" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{1+1}\\left(θ\\right)$$" ], "result": "=2\\sec^{1+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7aY4rYekMZhUfP6sZ6mfqd6Lsz0lt1XLEr88Kc/cjJo5V00rpv8+ZC6TM10tVCSHspqrCcTwb/I0f8OwMjuU6eBAUa+AHsHd5OGpMiuFITjOuqvW3hAHH635V/vEEYIatHimBRYRqHSWeJkuUPhfTC5LQ0B1JNgsb5SXzbS+SCnAfGB9H25Zc+Ua1z7Kw8kgb4gBJl4WMO1rA0a30/bUYlg==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "result": "=\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYs0Vsx3c0nt53Utu5DgHTe3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK00HV/bLwwglXz/2YNx2i4JHggSrmCoGjSdud4CyOvVOgdQN0k+N3NHNAkEIoLrhB631k80nn+ODwI4ID0kCF5+z7zRz81SqRD02HEUE4Wqh" } }, { "type": "step", "result": "=4\\left(2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Simplify $$4\\left(2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\right):{\\quad}4\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)$$", "input": "4\\left(2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\right)", "result": "=4\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "$$2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)=2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)$$", "input": "2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\tan\\left(θ\\right)\\tan\\left(θ\\right)=\\:\\tan^{1+1}\\left(θ\\right)$$" ], "result": "=2\\sec^{2}\\left(θ\\right)\\tan^{1+1}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Md1SJKHYLxhOYRRbsLTu+axUZgt6FTmFSHnRsotkqemLeJKmeakO1+Q3aSIldG41s7BwSVdKEYSqn+4fKAQoGk0xmFIl1KTO5aAkhaHvKzBALhShbMGZIZsZWTO721HsWI4dG63F/aSz/PFDvu5FnO3sngYNzcydIedGQjjtM8sBMUhkLMqDD6KHvgZAgKMdRQzIIgXxnWeDhDTk2Lfum8RPUSGriaFRn0NZXHZrPQw=" } }, { "type": "interim", "title": "$$\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)=\\sec^{4}\\left(θ\\right)$$", "input": "\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)=\\:\\sec^{2+2}\\left(θ\\right)$$" ], "result": "=\\sec^{2+2}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+2=4$$", "result": "=\\sec^{4}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7AtrsReczPdUqIdn5/Ev5mFtNfVJqszzpMwg6spHO8F8gJ/ZZA32ZInFBpDtxBfiKn3Jyj2VJgBojKQHjINpkkETYvnjqwhFu6aWmDOYpAVMO/U6qyRZonn1WPQ/21pz3cKXVLLQXMot2gtnZH7qY5zkbLioo7YVO0SRnDAO/7Uokt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "=4\\left(\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Simplify $$2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{4}\\left(θ\\right):{\\quad}3\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)$$", "input": "2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{4}\\left(θ\\right)", "result": "=4\\left(3\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Rewrite using trig identities", "input": "\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)", "result": "=-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Use the Pythagorean identity: $$\\tan^{2}\\left(x\\right)+1=\\sec^{2}\\left(x\\right)$$", "secondary": [ "$$\\tan^{2}\\left(x\\right)=\\sec^{2}\\left(x\\right)-1$$" ], "result": "=\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right)" }, { "type": "interim", "title": "Simplify $$\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right):{\\quad}3\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)$$", "input": "\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right)", "result": "=3\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "interim", "title": "Expand $$2\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right):{\\quad}2\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)$$", "input": "2\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right)", "result": "=\\sec^{4}\\left(θ\\right)+2\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b-c\\right)=ab-ac$$", "secondary": [ "$$a=2\\sec^{2}\\left(θ\\right),\\:b=\\sec^{2}\\left(θ\\right),\\:c=1$$" ], "result": "=2\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)\\cdot\\:1", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "result": "=2\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)-2\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$2\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)-2\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right):{\\quad}2\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)$$", "input": "2\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)-2\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right)", "result": "=2\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "interim", "title": "$$2\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)=2\\sec^{4}\\left(θ\\right)$$", "input": "2\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)=\\:\\sec^{2+2}\\left(θ\\right)$$" ], "result": "=2\\sec^{2+2}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+2=4$$", "result": "=2\\sec^{4}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Md1SJKHYLxhOYRRbsLTu+dJYPDPDesonRI520G8zYRPehkKrn0era9rz8TlL+x/vyvn14PieOyuPP98CkiPIaXRfFgXB+vlRb/AHCJklpF+qQC22PXYzQL8KzAhJCzszqX/cSuhK5Ty2xqd/IdNr67A2LXxYB8f3C3KpWaLgll1U2fg2Eo8khoQfnK/P3ock" } }, { "type": "interim", "title": "$$2\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right)=2\\sec^{2}\\left(θ\\right)$$", "input": "2\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=2\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ht1jgkupea8+jUgT+M/VFPgkIqVBCzsTrV5i/ECNCw8tOtZYwUjyXhDTsNnn6Elr8U8lPA4O6WXezAN1J2P7Ue6vfsn/ogOG+fHMfNQh9Je7opJQAzFYPpqn/2bLavXutzgLeG7CLgj8aHpc44iMEx91+u64Nzpp2/441mWGcQ/ET1Ehq4mhUZ9DWVx2az0M" } }, { "type": "step", "result": "=2\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Md1SJKHYLxhOYRRbsLTu+fjbMbEoNNbIR/7Q84OynCfTLx8mOdHYVzxX643JqKFI1Jetp9AvoQcY2QSDwFJhz1kUvGSawlk7umQ8kurx5doE3SiHEhDlxIXJDVY+PPdvgQUxJPyUNnGfVirkcwpVO6RMkE3rByM/Y1Dxxfs3xFw6FFV/0Hm6Y1hI7VaPzzKhkSoZP7CyArJlOo0FRny79Q==" } }, { "type": "step", "primary": "Add similar elements: $$\\sec^{4}\\left(θ\\right)+2\\sec^{4}\\left(θ\\right)=3\\sec^{4}\\left(θ\\right)$$", "result": "=3\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ARkhlGZdazrdxNLzhlqePtRiXXTzenxbt00ahL9sDS5L0MjjUe01mdeSwRMoYPSBLTrWWMFI8l4Q07DZ5+hJazaxvejhvTJvqoS105+T9+xc3qH/hVuCay9igw/1IwZBNySLUGdPDJeW3uY1XSs479bA+zX4bD3u3gx65o2NJhP8KqIsxz0lwScBMkZS31XhRCRwG0yFaMyhS3uC5/uUeMvqafR00cq/gMXQBnztqPn0yTzBsNnlyCSr/zrcxdQG" } } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7FkSOkw7Nc+ISjlBZ7XiESO/dTLaj0YrNSw6+/ogyGX8k+jBZ4XxujMMeS3Qvsi6m7aEe0hvT9VtxXei653t2zQi9Vnn4imguIoWn3N1eqjdGIkb3QW2w7y4N83RSktbc/kUupyp7xXxB8YmKEQ5HoPs4SAUa1zTfp2gkDWnxoYVtNtNgAtA4xHxju0w39zFkZEt3ZXAiqUE0HIXrrrezJJi0cdkgnw2HPjRVpcHMxZbIciYVzjaAolEaNkVONtBBRppTNgMFons9bMkTzIHooA==" } } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7aTuTexD/CTl68iqGJang3fkdSragnZ5K8+WMRM5qjJ6ETkUamdUz59t0aHpV3uN0rE0Ib0rKQ2oM97vl/gDbowCWKUbvV6WK3fDUgFtg3Q8XyqIsLZjT7Df8hUObtCZb+5k7zGGz9wmA2lyQsVWy+j6F2h9EjPuS9/S594y1Nd1FKk3fejFkyiOiq9iG9IkAIuSmJUDtLxRPyJ57Nkq7qlZOK6n0rXJ0Nc64aGIzHiSsVGYLehU5hUh50bKLZKnpo1d4a+pBCGznOTLxP2sni3enTnyZLeuZs6pBwuQMplUkt3WiGR7ZaCaXvz77bMjS" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)=12\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=12\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the Product Rule: $$\\left(f{\\cdot}g\\right)'=f'{\\cdot}g+f{\\cdot}g'$$", "secondary": [ "$$f=\\sec^{4}\\left(θ\\right),\\:g=\\tan\\left(θ\\right)$$" ], "result": "=12\\left(\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)\\tan\\left(θ\\right)+\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)\\sec^{4}\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Product%20Rule", "practiceTopic": "Product Rule" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)=4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{4},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{4}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{4}\\right)=4u^{3}$$", "input": "\\frac{d}{du}\\left(u^{4}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=4u^{4-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=4u^{3}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYl3oEe4EC6n4B8Xgz0cKnkqk3hxk9aCfAWodBRxXgUexuzo0Fl6VhneutEteidn+Kv8//6/nV5O4fb8Xgwi7mapNFxUvwBeni+JEIFAdbegzYEOcQgyW4n51cR7AeMAm7Q==" } }, { "type": "step", "result": "=4u^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgA2CO5cm49xNEqFV0DMaZJRqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidVduWEnoDYDg06IbaTsp9TFtqiuNhFTvmF8ZCDqWZNoubfnOah8gAvLMKOeNMctC5tVYZ/3r82LxPimKUFXjQFStvD8GKDNJJN8qiu0XRcVZAYAvbRKqkRVDaDBHkSBtDwYHkfmHHTze3JXk3LjiJp" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$4\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "4\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{3+1}\\left(θ\\right)$$" ], "result": "=4\\sec^{3+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$3+1=4$$", "result": "=4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7p9w76+/nvHitDrMIupHrrrPmAPxuc/RFHWXkB8A+mnqLeJKmeakO1+Q3aSIldG41PCyA7ixjCdfDYe+yac7KOvOIrkYdKW3Q58EnXaMU+7oslHPSvPf5UfEs5T+IFZ4XHjb2+5NLFZrsH9fcPWg/TXL71sHPAsAQtYZvjpaGSRwboK/Igc1IRC3EGruCoRzV1Zrc/SYRIR8JuxYpD+PJ+Q==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "result": "=\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYs0Vsx3c0nt53Utu5DgHTe3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK00HV/bLwwglXz/2YNx2i4JHggSrmCoGjSdud4CyOvVOgdQN0k+N3NHNAkEIoLrhB631k80nn+ODwI4ID0kCF5+z7zRz81SqRD02HEUE4Wqh" } }, { "type": "step", "result": "=12\\left(4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Simplify $$12\\left(4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right):{\\quad}12\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)$$", "input": "12\\left(4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right)", "result": "=12\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "$$4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)=4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)$$", "input": "4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\tan\\left(θ\\right)\\tan\\left(θ\\right)=\\:\\tan^{1+1}\\left(θ\\right)$$" ], "result": "=4\\sec^{4}\\left(θ\\right)\\tan^{1+1}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7GHerUg4zB8o0EeBO6BKV16xUZgt6FTmFSHnRsotkqemLeJKmeakO1+Q3aSIldG41PCyA7ixjCdfDYe+yac7KOvOIrkYdKW3Q58EnXaMU+7pALhShbMGZIZsZWTO721HsXaBT4KNKuhZZSFsFWnKRStdNJOkRdCtrWB4iMQVmMwBoAWE3NiWlLPrDNnXLOZLK3tKI9zGyi4xCTaccmboyHcRPUSGriaFRn0NZXHZrPQw=" } }, { "type": "interim", "title": "$$\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)=\\sec^{6}\\left(θ\\right)$$", "input": "\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)=\\:\\sec^{2+4}\\left(θ\\right)$$" ], "result": "=\\sec^{2+4}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+4=6$$", "result": "=\\sec^{6}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7AtrsReczPdUqIdn5/Ev5mLBxohAhN47qWD2nlwOHqfkgJ/ZZA32ZInFBpDtxBfiKn3Jyj2VJgBojKQHjINpkkG6JHhxn2McSW7OxLgQBi64O/U6qyRZonn1WPQ/21pz3xMe3sR8ZMoA2Ynlp2W853t3OcQiSW6rZ4zI81UhADI4kt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "=12\\left(\\sec^{6}\\left(θ\\right)+4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7lv52tOJZUIL6vqIybdOpThiMJmjxigW8LortZKvrqEbWPF6oWuXggupsFvXrooIqxMe3sR8ZMoA2Ynlp2W853tMvHyY50dhXPFfrjcmooUinnaTA6rKFuYRlEeyjdJSquO1NenZJP60hEwBH05kmlvOIrkYdKW3Q58EnXaMU+7r4ZAjrJM4mwvgfYovsxoopgQUxJPyUNnGfVirkcwpVO48SyOdzKmRYNaCIczIhnoIh0PvFonSnuZi7Z6ItIhmYhYbzXwn7iok10o2w+NJLRiZIb9CFuBJdPyi+MBoLCMd/LnYygZMYhjzpUCXiHwCy" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=8\\left(-4\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)+12\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(24\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)\\right)=24\\left(64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+132\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+14\\sec^{8}\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(24\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=24\\frac{d}{dθ}\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=24\\left(\\frac{d}{dθ}\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)\\right)+\\frac{d}{dθ}\\left(14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)\\right)=16\\left(4\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+3\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=16\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the Product Rule: $$\\left(f{\\cdot}g\\right)'=f'{\\cdot}g+f{\\cdot}g'$$", "secondary": [ "$$f=\\sec^{4}\\left(θ\\right),\\:g=\\tan^{3}\\left(θ\\right)$$" ], "result": "=16\\left(\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)\\tan^{3}\\left(θ\\right)+\\frac{d}{dθ}\\left(\\tan^{3}\\left(θ\\right)\\right)\\sec^{4}\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Product%20Rule", "practiceTopic": "Product Rule" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)=4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{4},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{4}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{4}\\right)=4u^{3}$$", "input": "\\frac{d}{du}\\left(u^{4}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=4u^{4-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=4u^{3}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYl3oEe4EC6n4B8Xgz0cKnkqk3hxk9aCfAWodBRxXgUexuzo0Fl6VhneutEteidn+Kv8//6/nV5O4fb8Xgwi7mapNFxUvwBeni+JEIFAdbegzYEOcQgyW4n51cR7AeMAm7Q==" } }, { "type": "step", "result": "=4u^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgA2CO5cm49xNEqFV0DMaZJRqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidVduWEnoDYDg06IbaTsp9TFtqiuNhFTvmF8ZCDqWZNoubfnOah8gAvLMKOeNMctC5tVYZ/3r82LxPimKUFXjQFStvD8GKDNJJN8qiu0XRcVZAYAvbRKqkRVDaDBHkSBtDwYHkfmHHTze3JXk3LjiJp" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$4\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "4\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{3+1}\\left(θ\\right)$$" ], "result": "=4\\sec^{3+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$3+1=4$$", "result": "=4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7p9w76+/nvHitDrMIupHrrrPmAPxuc/RFHWXkB8A+mnqLeJKmeakO1+Q3aSIldG41PCyA7ixjCdfDYe+yac7KOvOIrkYdKW3Q58EnXaMU+7oslHPSvPf5UfEs5T+IFZ4XHjb2+5NLFZrsH9fcPWg/TXL71sHPAsAQtYZvjpaGSRwboK/Igc1IRC3EGruCoRzV1Zrc/SYRIR8JuxYpD+PJ+Q==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan^{3}\\left(θ\\right)\\right)=3\\tan^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan^{3}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}3\\left(\\tan\\left(θ\\right)\\right)^{2}\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan^{3}\\left(θ\\right)\\right)", "result": "=3\\left(\\tan\\left(θ\\right)\\right)^{2}\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{3},\\:\\:u=\\tan\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{3}\\right)\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{3}\\right)=3u^{2}$$", "input": "\\frac{d}{du}\\left(u^{3}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=3u^{3-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=3u^{2}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYjNrXZy15fg+DDm4IZ/kmJKk3hxk9aCfAWodBRxXgUexYrhQEWwJJMG//0JI2CMZRv8//6/nV5O4fb8Xgwi7mapNFxUvwBeni+JEIFAdbegz8nAGF3XMCO2i5cp0uYL6iA==" } }, { "type": "step", "result": "=3u^{2}\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\tan\\left(θ\\right)$$", "result": "=3\\left(\\tan\\left(θ\\right)\\right)^{2}\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYux7TmDJ3fopnHGBxoqO+pYseUrH1bTMvpsf9SfQ9YpVqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKifDI4IHJ+buH5M+0PHLeGB4Be4N1NohQPDRW/BmsNJaPsq1KfCFKwSwA0qmWovUVP9tVYZ/3r82LxPimKUFXjQFStvD8GKDNJJN8qiu0XRcVZAYAvbRKqkRVDaDBHkSBtDwYHkfmHHTze3JXk3LjiJp" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "result": "=\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYs0Vsx3c0nt53Utu5DgHTe3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK00HV/bLwwglXz/2YNx2i4JHggSrmCoGjSdud4CyOvVOgdQN0k+N3NHNAkEIoLrhB631k80nn+ODwI4ID0kCF5+z7zRz81SqRD02HEUE4Wqh" } }, { "type": "step", "result": "=3\\left(\\tan\\left(θ\\right)\\right)^{2}\\sec^{2}\\left(θ\\right)" }, { "type": "step", "primary": "Simplify", "result": "=3\\tan^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=16\\left(4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{3}\\left(θ\\right)+3\\tan^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Simplify $$16\\left(4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{3}\\left(θ\\right)+3\\tan^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right):{\\quad}16\\left(4\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+3\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)$$", "input": "16\\left(4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{3}\\left(θ\\right)+3\\tan^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right)", "result": "=16\\left(4\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+3\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "$$4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{3}\\left(θ\\right)=4\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)$$", "input": "4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{3}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\tan\\left(θ\\right)\\tan^{3}\\left(θ\\right)=\\:\\tan^{1+3}\\left(θ\\right)$$" ], "result": "=4\\sec^{4}\\left(θ\\right)\\tan^{1+3}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+3=4$$", "result": "=4\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7GHerUg4zB8o0EeBO6BKV162cmBc6Js8szDUhC2HxdqjUgkMhq/lXB9qI3pCS5SJMq47vuWedXv2WUg94ER8IwVADhNIEz39BpXewdhMCBb8aab3UY3H06dpMJimNIyD5kuHx4ulZqfxZHIsiQUpV1nyo9elyUFFpcPZW02cAt7EdWYTaqPpaDtAHPW/SgMNicUwY1HWd9EJWOCb4O+kauAND0fAz408eCOOnBAkZ69Akt3WiGR7ZaCaXvz77bMjS" } }, { "type": "interim", "title": "$$3\\tan^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)=3\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)$$", "input": "3\\tan^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)=\\:\\sec^{2+4}\\left(θ\\right)$$" ], "result": "=3\\tan^{2}\\left(θ\\right)\\sec^{2+4}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+4=6$$", "result": "=3\\tan^{2}\\left(θ\\right)\\sec^{6}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7xud7SC9MT1MQdbTGBqrMw+H0qCTik4j9uS7jTF7ng2/8DCHl2/1cUFH2kQZvGLuVzRqDxPUzBN6vjj5oJL9kUEngkySEzEl6diiSTBzPnLnWxcLMxZwB9cGBLGbwbyjTrqr1t4QBx+t+Vf7xBGCGrW9B4jZ4XSgntIkqjxqF2Bvh9Kgk4pOI/bku40xe54Nv6RICywOwLOWusKkGpuhUtYx6/teVLREoXphSPTLZ2MbET1Ehq4mhUZ9DWVx2az0M" } }, { "type": "step", "result": "=16\\left(3\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+4\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7C43rUQmr5KAYI3UGEng6yRiMJmjxigW8LortZKvrqEYERpCXXFEE4hLCZfTeroF6V7JvkhZ2EFhPsvL8iYbkgPpadcGORE5bHX37Io3viWolp/MB1OadflBmLQoq5Pmlo5FYteSPKwXny4uCMrdsKwS9KcmQjztCcdjD5gRjJOz1a03PQ0UrDYBem+/aln9X84iuRh0pbdDnwSddoxT7uh96fmgcT2t23FrnniYZ1CmBBTEk/JQ2cZ9WKuRzClU7UxxUEaCmP0MbiSArdWRj6iHQ+8WidKe5mLtnoi0iGZhpK7ceo/th4B1fP0TRnZr9A1c/UDX6QxYlZyg/KYfMkDSei4QZxxrazp/cx7cYZvYA0hWLB/nt1m4ggrh+mxlbvzIPeEtDfcHv/z8uls8Teg==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)=14\\left(6\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{8}\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=14\\frac{d}{dθ}\\left(\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the Product Rule: $$\\left(f{\\cdot}g\\right)'=f'{\\cdot}g+f{\\cdot}g'$$", "secondary": [ "$$f=\\sec^{6}\\left(θ\\right),\\:g=\\tan\\left(θ\\right)$$" ], "result": "=14\\left(\\frac{d}{dθ}\\left(\\sec^{6}\\left(θ\\right)\\right)\\tan\\left(θ\\right)+\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)\\sec^{6}\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Product%20Rule", "practiceTopic": "Product Rule" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec^{6}\\left(θ\\right)\\right)=6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{6}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}6\\left(\\sec\\left(θ\\right)\\right)^{5}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{6}\\left(θ\\right)\\right)", "result": "=6\\left(\\sec\\left(θ\\right)\\right)^{5}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{6},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{6}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{6}\\right)=6u^{5}$$", "input": "\\frac{d}{du}\\left(u^{6}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=6u^{6-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=6u^{5}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYgOK76JOSNqGa1B1wU6ozWWk3hxk9aCfAWodBRxXgUexqSoDkuZ4Ab1St3bM8oqKcP8//6/nV5O4fb8Xgwi7mapNFxUvwBeni+JEIFAdbegzHfwjUTnSu4cWqw+8m79+NA==" } }, { "type": "step", "result": "=6u^{5}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=6\\left(\\sec\\left(θ\\right)\\right)^{5}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgBaJTGF5uPwSMzxoSyJNVZsqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKieMSe35zYqdvyntnEYxaQ88jJq7JZFtKgvxzzzuaqTGrebfnOah8gAvLMKOeNMctC5tVYZ/3r82LxPimKUFXjQFStvD8GKDNJJN8qiu0XRcVZAYAvbRKqkRVDaDBHkSBtDwYHkfmHHTze3JXk3LjiJp" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=6\\left(\\sec\\left(θ\\right)\\right)^{5}\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$6\\sec^{5}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "6\\sec^{5}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{5}\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{5+1}\\left(θ\\right)$$" ], "result": "=6\\sec^{5+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$5+1=6$$", "result": "=6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7QMJmgA2Sx2hYpGS5j/h1NLPmAPxuc/RFHWXkB8A+mnqLeJKmeakO1+Q3aSIldG417JiujyaxykLR7Tv0Nr+wOGprXgPFMmP3i5vY2oD+e4YslHPSvPf5UfEs5T+IFZ4XHjb2+5NLFZrsH9fcPWg/TRiIlnZT/wCWlhB0uSVNtvx/uR/EZqc6EbI3tnxSI5001Zrc/SYRIR8JuxYpD+PJ+Q==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "result": "=\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYs0Vsx3c0nt53Utu5DgHTe3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK00HV/bLwwglXz/2YNx2i4JHggSrmCoGjSdud4CyOvVOgdQN0k+N3NHNAkEIoLrhB631k80nn+ODwI4ID0kCF5+z7zRz81SqRD02HEUE4Wqh" } }, { "type": "step", "result": "=14\\left(6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{6}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Simplify $$14\\left(6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{6}\\left(θ\\right)\\right):{\\quad}14\\left(6\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{8}\\left(θ\\right)\\right)$$", "input": "14\\left(6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{6}\\left(θ\\right)\\right)", "result": "=14\\left(6\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{8}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "$$6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)=6\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)$$", "input": "6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\tan\\left(θ\\right)\\tan\\left(θ\\right)=\\:\\tan^{1+1}\\left(θ\\right)$$" ], "result": "=6\\sec^{6}\\left(θ\\right)\\tan^{1+1}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=6\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7H4jwT8QhunkiDAGNb+RcvaxUZgt6FTmFSHnRsotkqemLeJKmeakO1+Q3aSIldG417JiujyaxykLR7Tv0Nr+wOGprXgPFMmP3i5vY2oD+e4ZALhShbMGZIZsZWTO721Hsn1G20l7DmbyzInDIeLRU/ZYojigZ8i1ymB3KwWBswnjgWFmp7EeEDyNH2q8Tm1k1jHr+15UtEShemFI9MtnYxsRPUSGriaFRn0NZXHZrPQw=" } }, { "type": "interim", "title": "$$\\sec^{2}\\left(θ\\right)\\sec^{6}\\left(θ\\right)=\\sec^{8}\\left(θ\\right)$$", "input": "\\sec^{2}\\left(θ\\right)\\sec^{6}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{2}\\left(θ\\right)\\sec^{6}\\left(θ\\right)=\\:\\sec^{2+6}\\left(θ\\right)$$" ], "result": "=\\sec^{2+6}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+6=8$$", "result": "=\\sec^{8}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7AtrsReczPdUqIdn5/Ev5mH+O3FtTTH45VMxFH8DJ4BEgJ/ZZA32ZInFBpDtxBfiKn3Jyj2VJgBojKQHjINpkkI6d/oVrekw7T/Cky7x1KxcO/U6qyRZonn1WPQ/21pz3HK+miBBOGkRkZzIN3nn5xFhBmsSvLw1yh3wbSiSoUDgkt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "=14\\left(\\sec^{8}\\left(θ\\right)+6\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7nO0mFYRqkI6v7Sy9VTWwjRiMJmjxigW8LortZKvrqEbWPF6oWuXggupsFvXrooIqHK+miBBOGkRkZzIN3nn5xNMvHyY50dhXPFfrjcmooUiPqGSxzi9gRanHsLmpQgKwsfoApmSO56Ak/1ybzTsFLmprXgPFMmP3i5vY2oD+e4b4ZAjrJM4mwvgfYovsxoopgQUxJPyUNnGfVirkcwpVO2TUlQSCHNJw/4wJ8m9ormYqoLNuWbSjE1JT+hrPzX/RhYbzXwn7iok10o2w+NJLRiZIb9CFuBJdPyi+MBoLCMeBkcS/lgaNKtB6zcW1E3BB" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=24\\left(16\\left(4\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+3\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)+14\\left(6\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{8}\\left(θ\\right)\\right)\\right)" }, { "type": "interim", "title": "Expand $$16\\left(4\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+3\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)+14\\left(6\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{8}\\left(θ\\right)\\right):{\\quad}64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+132\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+14\\sec^{8}\\left(θ\\right)$$", "input": "16\\left(4\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+3\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)+14\\left(6\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{8}\\left(θ\\right)\\right)", "result": "=24\\left(64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+132\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+14\\sec^{8}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Expand $$16\\left(4\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+3\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right):{\\quad}64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+48\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)$$", "input": "16\\left(4\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+3\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)", "result": "=64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+48\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+14\\left(6\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{8}\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b+c\\right)=ab+ac$$", "secondary": [ "$$a=16,\\:b=4\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right),\\:c=3\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)$$" ], "result": "=16\\cdot\\:4\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+16\\cdot\\:3\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "interim", "title": "Simplify $$16\\cdot\\:4\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+16\\cdot\\:3\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right):{\\quad}64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+48\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)$$", "input": "16\\cdot\\:4\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+16\\cdot\\:3\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)", "result": "=64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+48\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$16\\cdot\\:4=64$$", "result": "=64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+16\\cdot\\:3\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)" }, { "type": "step", "primary": "Multiply the numbers: $$16\\cdot\\:3=48$$", "result": "=64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+48\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7C43rUQmr5KAYI3UGEng6yV6kSs12eASfWmxynRkYZ3imWJLfEdyBmZh9L6HIlMnFSPKgibEOx6xAH14rsVQdDd6GQqufR6tr2vPxOUv7H+/wV+6wolBQOY9LPugwYJ6C3tKI9zGyi4xCTaccmboyHVWGy3JBiLa3A/i0ciCokDrRsdR53zGMNMouUmBvz53xupxJf7boNv1S9Hc0DXJXvt7UD16RqFQha281as3i/3gIhVb1SjznyQ5krYRiFqNmoU0M/hfYOSS/FiVCoXaNrXxGVgveSDjBoBB0CLM8z96ArlFiGY8qAdw44hDradHvvzIPeEtDfcHv/z8uls8Teg==" } }, { "type": "interim", "title": "Expand $$14\\left(6\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{8}\\left(θ\\right)\\right):{\\quad}84\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+14\\sec^{8}\\left(θ\\right)$$", "input": "14\\left(6\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{8}\\left(θ\\right)\\right)", "result": "=64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+48\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+84\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+14\\sec^{8}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b+c\\right)=ab+ac$$", "secondary": [ "$$a=14,\\:b=6\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right),\\:c=\\sec^{8}\\left(θ\\right)$$" ], "result": "=14\\cdot\\:6\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+14\\sec^{8}\\left(θ\\right)", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "primary": "Multiply the numbers: $$14\\cdot\\:6=84$$", "result": "=84\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+14\\sec^{8}\\left(θ\\right)" } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7nO0mFYRqkI6v7Sy9VTWwjSvxearhctbAPxX3QGBh5kkx0tWj1vW7tIvP14Mwoy6mLTrWWMFI8l4Q07DZ5+hJazPkBMIuMrWSeavgWBoF37pgEq4U4c+VEIUu/cb6SEfLgBHHd6NlsvdNQCpoDejgKWxRFK5IwscCsNiX9YX0G6d6pfF1z6umzUJTJvt+ojYZ00PxkCV/xFqV0mmm+Nwrth1Y04xtNK2BCM2tE3PxmDme5tn70JIT2jKyMskERPvitGNYJGuT8kFILfu1szjq+A==" } }, { "type": "step", "primary": "Add similar elements: $$48\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+84\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)=132\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)$$", "result": "=64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+132\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+14\\sec^{8}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=-8\\left(-4\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)+12\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)\\right)+24\\left(64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+132\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+14\\sec^{8}\\left(θ\\right)\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{d}{dθ}\\left(-8\\left(-4\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)+12\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)\\right)+24\\left(64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+132\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+14\\sec^{8}\\left(θ\\right)\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(-8\\left(-4\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)+12\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)\\right)+24\\left(64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+132\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+14\\sec^{8}\\left(θ\\right)\\right)\\right)=-8\\left(-4\\left(-4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+12\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)\\right)+24\\left(256\\tan^{5}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+1048\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+376\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(-8\\left(-4\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)+12\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)\\right)+24\\left(64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+132\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+14\\sec^{8}\\left(θ\\right)\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=-\\frac{d}{dθ}\\left(8\\left(-4\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)+12\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)\\right)\\right)+\\frac{d}{dθ}\\left(24\\left(64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+132\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+14\\sec^{8}\\left(θ\\right)\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(8\\left(-4\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)+12\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)\\right)\\right)=8\\left(-4\\left(-4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+12\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(8\\left(-4\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)+12\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=8\\frac{d}{dθ}\\left(-4\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)+12\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)\\right)" }, { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=8\\left(-\\frac{d}{dθ}\\left(4\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)\\right)+\\frac{d}{dθ}\\left(12\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(4\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)\\right)=4\\left(-4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(4\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=4\\frac{d}{dθ}\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=4\\left(-\\frac{d}{dθ}\\left(2\\sec^{2}\\left(θ\\right)\\right)+\\frac{d}{dθ}\\left(3\\sec^{4}\\left(θ\\right)\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(2\\sec^{2}\\left(θ\\right)\\right)=4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(2\\sec^{2}\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=2\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Apply the chain rule:$${\\quad}2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{2},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{2}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{2}\\right)=2u$$", "input": "\\frac{d}{du}\\left(u^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2u^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2u", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYr+VZUwBnLdzbS6DZQ+f4s+k3hxk9aCfAWodBRxXgUexMchyqTAJWrzJaDbnNcFsJUeCBKuYKgaNJ253gLI69U79qbCA2QqVmvm3jGRXZ2ppvbGT4j1utMEkCDH25m/vlQ==" } }, { "type": "step", "result": "=2u\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgCSXIThcWH58oYoMPt18lzFqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidZClirlWNFPnHR1rvdyoFLI7IC+oqnwb1T6nPLpiw1eysw2NXPd8jZnr3fcPpqc7ejeh7+jKEzLb7VNCEMF3Z/Maswa5oIagKQFOfrnKm5+Gbb/PxX3awut5KxRxyZf8A=" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=2\\cdot\\:2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$2\\cdot\\:2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "2\\cdot\\:2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=4\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{1+1}\\left(θ\\right)$$" ], "result": "=4\\sec^{1+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s71jEHNygbGF2qQyJrTxQFvpK/zREn/U8meKQ3TvRjTQz8cZMsH3IX6lxp+ANHGbuCq47vuWedXv2WUg94ER8IwfulLd9uW7xFGDkqRxB3vSSe0Rv083m3K3ie3DVArOu08LfSxJ+0AgVLpCSnLX0iSnTHyKtbMq++mkoxS+pOuesyFZaieHAvmtaw5r6RKguDHxgfR9uWXPlGtc+ysPJIG+IASZeFjDtawNGt9P21GJY=" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(3\\sec^{4}\\left(θ\\right)\\right)=12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(3\\sec^{4}\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=3\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Apply the chain rule:$${\\quad}4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{4},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{4}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{4}\\right)=4u^{3}$$", "input": "\\frac{d}{du}\\left(u^{4}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=4u^{4-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=4u^{3}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYl3oEe4EC6n4B8Xgz0cKnkqk3hxk9aCfAWodBRxXgUexuzo0Fl6VhneutEteidn+Kv8//6/nV5O4fb8Xgwi7mapNFxUvwBeni+JEIFAdbegzYEOcQgyW4n51cR7AeMAm7Q==" } }, { "type": "step", "result": "=4u^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgA2CO5cm49xNEqFV0DMaZJRqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidVduWEnoDYDg06IbaTsp9TFtqiuNhFTvmF8ZCDqWZNoubfnOah8gAvLMKOeNMctC5tVYZ/3r82LxPimKUFXjQFStvD8GKDNJJN8qiu0XRcVZAYAvbRKqkRVDaDBHkSBtDwYHkfmHHTze3JXk3LjiJp" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=3\\cdot\\:4\\left(\\sec\\left(θ\\right)\\right)^{3}\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$3\\cdot\\:4\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "3\\cdot\\:4\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$3\\cdot\\:4=12$$", "result": "=12\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{3+1}\\left(θ\\right)$$" ], "result": "=12\\sec^{3+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$3+1=4$$", "result": "=12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7jFGb0q9vd+kdWvVYeHJHbDNSoP5XQt+n/5y371e8QHJROlTYN926lBwmiJdD5HUSzRqDxPUzBN6vjj5oJL9kUFPzsE52Mw7P00gxEE61byKp9ej8WEAztP3UNEC9n7jjZEt3ZXAiqUE0HIXrrrezJHZKl2mV4AlztiApTAFigz2gNkWvAT/c3O3uDO/qxcKwM1Kg/ldC36f/nLfvV7xActytFlzCV8oq55bHLwXbb0c=" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=4\\left(-4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(12\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)\\right)=12\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(12\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=12\\frac{d}{dθ}\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=12\\left(\\frac{d}{dθ}\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)+\\frac{d}{dθ}\\left(\\sec^{6}\\left(θ\\right)\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)=4\\left(4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=4\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the Product Rule: $$\\left(f{\\cdot}g\\right)'=f'{\\cdot}g+f{\\cdot}g'$$", "secondary": [ "$$f=\\sec^{4}\\left(θ\\right),\\:g=\\tan^{2}\\left(θ\\right)$$" ], "result": "=4\\left(\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)\\tan^{2}\\left(θ\\right)+\\frac{d}{dθ}\\left(\\tan^{2}\\left(θ\\right)\\right)\\sec^{4}\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Product%20Rule", "practiceTopic": "Product Rule" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)=4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{4},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{4}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{4}\\right)=4u^{3}$$", "input": "\\frac{d}{du}\\left(u^{4}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=4u^{4-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=4u^{3}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYl3oEe4EC6n4B8Xgz0cKnkqk3hxk9aCfAWodBRxXgUexuzo0Fl6VhneutEteidn+Kv8//6/nV5O4fb8Xgwi7mapNFxUvwBeni+JEIFAdbegzYEOcQgyW4n51cR7AeMAm7Q==" } }, { "type": "step", "result": "=4u^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgA2CO5cm49xNEqFV0DMaZJRqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidVduWEnoDYDg06IbaTsp9TFtqiuNhFTvmF8ZCDqWZNoubfnOah8gAvLMKOeNMctC5tVYZ/3r82LxPimKUFXjQFStvD8GKDNJJN8qiu0XRcVZAYAvbRKqkRVDaDBHkSBtDwYHkfmHHTze3JXk3LjiJp" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$4\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "4\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{3+1}\\left(θ\\right)$$" ], "result": "=4\\sec^{3+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$3+1=4$$", "result": "=4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7p9w76+/nvHitDrMIupHrrrPmAPxuc/RFHWXkB8A+mnqLeJKmeakO1+Q3aSIldG41PCyA7ixjCdfDYe+yac7KOvOIrkYdKW3Q58EnXaMU+7oslHPSvPf5UfEs5T+IFZ4XHjb2+5NLFZrsH9fcPWg/TXL71sHPAsAQtYZvjpaGSRwboK/Igc1IRC3EGruCoRzV1Zrc/SYRIR8JuxYpD+PJ+Q==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan^{2}\\left(θ\\right)\\right)=2\\tan\\left(θ\\right)\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan^{2}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}2\\tan\\left(θ\\right)\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan^{2}\\left(θ\\right)\\right)", "result": "=2\\tan\\left(θ\\right)\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{2},\\:\\:u=\\tan\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{2}\\right)\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{2}\\right)=2u$$", "input": "\\frac{d}{du}\\left(u^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2u^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2u", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYr+VZUwBnLdzbS6DZQ+f4s+k3hxk9aCfAWodBRxXgUexMchyqTAJWrzJaDbnNcFsJUeCBKuYKgaNJ253gLI69U79qbCA2QqVmvm3jGRXZ2ppvbGT4j1utMEkCDH25m/vlQ==" } }, { "type": "step", "result": "=2u\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\tan\\left(θ\\right)$$", "result": "=2\\tan\\left(θ\\right)\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYux7TmDJ3fopnHGBxoqO+paSXIThcWH58oYoMPt18lzFqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidIn+HfeW3QQwAkRa15J0NSAlj501WLIv46IdbBaParYow2V0WdtbiweShXv1KOwqCjeh7+jKEzLb7VNCEMF3Z/Maswa5oIagKQFOfrnKm5+Gbb/PxX3awut5KxRxyZf8A=" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "result": "=\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYs0Vsx3c0nt53Utu5DgHTe3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK00HV/bLwwglXz/2YNx2i4JHggSrmCoGjSdud4CyOvVOgdQN0k+N3NHNAkEIoLrhB631k80nn+ODwI4ID0kCF5+z7zRz81SqRD02HEUE4Wqh" } }, { "type": "step", "result": "=2\\tan\\left(θ\\right)\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=4\\left(4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{2}\\left(θ\\right)+2\\tan\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Simplify $$4\\left(4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{2}\\left(θ\\right)+2\\tan\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right):{\\quad}4\\left(4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)$$", "input": "4\\left(4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{2}\\left(θ\\right)+2\\tan\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right)", "result": "=4\\left(4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "$$4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{2}\\left(θ\\right)=4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)$$", "input": "4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\tan\\left(θ\\right)\\tan^{2}\\left(θ\\right)=\\:\\tan^{1+2}\\left(θ\\right)$$" ], "result": "=4\\sec^{4}\\left(θ\\right)\\tan^{1+2}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+2=3$$", "result": "=4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7GHerUg4zB8o0EeBO6BKV162cmBc6Js8szDUhC2Hxdqi5HNbiQL8+p2P+VJTsJbU7q47vuWedXv2WUg94ER8IwVADhNIEz39BpXewdhMCBb+bbDkw/o6UIkLjzolgTAsqkuHx4ulZqfxZHIsiQUpV1nyo9elyUFFpcPZW02cAt7EtbJdlgVP3iSyKFg6v1zm6cUwY1HWd9EJWOCb4O+kauMdM6yd/xAZrmYAkuRY8/rYkt3WiGR7ZaCaXvz77bMjS" } }, { "type": "interim", "title": "$$2\\tan\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)=2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "2\\tan\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)=\\:\\sec^{2+4}\\left(θ\\right)$$" ], "result": "=2\\tan\\left(θ\\right)\\sec^{2+4}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+4=6$$", "result": "=2\\tan\\left(θ\\right)\\sec^{6}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7OyotidIxGYYp2AND0rbhNaCGttAzkBWvgG18MKDLv3Rdbcltti6PzkTDSwkmBTY6q47vuWedXv2WUg94ER8IwSIYCdSNn6m2gNv8FkjgS1XBL9WNCz07kx2T4iGijNtQjT/+cTnIKvMqhzKaFBbxvzSei4QZxxrazp/cx7cYZvbVeYdTw2sAeZ9o0nz1uUihamteA8UyY/eLm9jagP57hkwLhsnE4D1C6IGVVz/pqlY=" } }, { "type": "step", "result": "=4\\left(2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)+4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7iza9NvP7nq2MIMeLHhROsvkdSragnZ5K8+WMRM5qjJ4/XhyePE1SthuGDJcaFMnncKXVLLQXMot2gtnZH7qY51d1u0XlM+NdfDNBhSESuvlV00rpv8+ZC6TM10tVCSHsi/RlR31DZv0142/jAcann2ASrhThz5UQhS79xvpIR8tIoCq5NivR6sCY2Sfa33kc8jzDW/n9HorFBpgomiqIe+5byrQDQVCXUD0vH/fvOdxyhd7tjiG+GxQNxDvGkZUlqqyy15Nuz77zqSegKeIT2iKI9Km/ZvzCnLF9eqmklKHqRJDJ/DeTzEayKJpmRlyWQi730E1b9JEo9956Qy3sPLLWww1IW5GUOlAGrcUL2OqwiNrEngO+NNvZ9sqNu+2V" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec^{6}\\left(θ\\right)\\right)=6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{6}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}6\\left(\\sec\\left(θ\\right)\\right)^{5}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{6}\\left(θ\\right)\\right)", "result": "=6\\left(\\sec\\left(θ\\right)\\right)^{5}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{6},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{6}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{6}\\right)=6u^{5}$$", "input": "\\frac{d}{du}\\left(u^{6}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=6u^{6-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=6u^{5}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYgOK76JOSNqGa1B1wU6ozWWk3hxk9aCfAWodBRxXgUexqSoDkuZ4Ab1St3bM8oqKcP8//6/nV5O4fb8Xgwi7mapNFxUvwBeni+JEIFAdbegzHfwjUTnSu4cWqw+8m79+NA==" } }, { "type": "step", "result": "=6u^{5}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=6\\left(\\sec\\left(θ\\right)\\right)^{5}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgBaJTGF5uPwSMzxoSyJNVZsqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKieMSe35zYqdvyntnEYxaQ88jJq7JZFtKgvxzzzuaqTGrebfnOah8gAvLMKOeNMctC5tVYZ/3r82LxPimKUFXjQFStvD8GKDNJJN8qiu0XRcVZAYAvbRKqkRVDaDBHkSBtDwYHkfmHHTze3JXk3LjiJp" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=6\\left(\\sec\\left(θ\\right)\\right)^{5}\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$6\\sec^{5}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "6\\sec^{5}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{5}\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{5+1}\\left(θ\\right)$$" ], "result": "=6\\sec^{5+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$5+1=6$$", "result": "=6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7QMJmgA2Sx2hYpGS5j/h1NLPmAPxuc/RFHWXkB8A+mnqLeJKmeakO1+Q3aSIldG417JiujyaxykLR7Tv0Nr+wOGprXgPFMmP3i5vY2oD+e4YslHPSvPf5UfEs5T+IFZ4XHjb2+5NLFZrsH9fcPWg/TRiIlnZT/wCWlhB0uSVNtvx/uR/EZqc6EbI3tnxSI5001Zrc/SYRIR8JuxYpD+PJ+Q==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=12\\left(4\\left(4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Expand $$4\\left(4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "4\\left(4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=12\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Expand $$4\\left(4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right):{\\quad}16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+8\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "4\\left(4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)", "result": "=16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+8\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)+6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b+c\\right)=ab+ac$$", "secondary": [ "$$a=4,\\:b=4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right),\\:c=2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)$$" ], "result": "=4\\cdot\\:4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+4\\cdot\\:2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "interim", "title": "Simplify $$4\\cdot\\:4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+4\\cdot\\:2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+8\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "4\\cdot\\:4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+4\\cdot\\:2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+8\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:4=16$$", "result": "=16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+4\\cdot\\:2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:2=8$$", "result": "=16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+8\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7iza9NvP7nq2MIMeLHhROsoL8UIMToMV6hDKJhY+Nk7toxrHhSJArgBpsnuAV9UCfBgOJ+lz5D5t+hsCNYCoDMHWD310L1+P2yDQQfMEhENFHx6CvpVa90MvMzklpzJWXZAzoThz1g+TITtf+WJ7KiyRRh7kCo0obc/CTaNyPGkYuuAKnMbBNnPJ2x/QM7ydV1sD7NfhsPe7eDHrmjY0mEyzoLVQPyp9R+5GC0prpwlf2K8DiYaJsjNsjOcdOcdQLW6BryIKCtddVLZsZyZ1dV9DmtpfPTl/o4ZJKfQIdsQqwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "primary": "Add similar elements: $$8\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)+6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)=14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7iza9NvP7nq2MIMeLHhROsoL8UIMToMV6hDKJhY+Nk7toxrHhSJArgBpsnuAV9UCf8SVGtBi1ovwgT5vDB2QfkF3B8WOxYrIgIdHvusvufLAAlilG71elit3w1IBbYN0PumuR+HimzGDTYbsfY1Q6JXyo9elyUFFpcPZW02cAt7EXYiLGBL7PG2xAqJd1w6BXKK1Q1tVX7SUdYWONKsZTX66q9beEAcfrflX+8QRghq2CcvL2jSzw1VlRcGX4ka6qwZ5KNwm43WeyBMBBKL30QvI8w1v5/R6KxQaYKJoqiHsLiDrnJdtlx1I2mhkkUTElN8b/k79QNO9t6sAoPBR3zJYojigZ8i1ymB3KwWBswnjiAEmXhYw7WsDRrfT9tRiW" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=8\\left(-4\\left(-4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+12\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(24\\left(64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+132\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+14\\sec^{8}\\left(θ\\right)\\right)\\right)=24\\left(256\\tan^{5}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+1048\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+376\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(24\\left(64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+132\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+14\\sec^{8}\\left(θ\\right)\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=24\\frac{d}{dθ}\\left(64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+132\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+14\\sec^{8}\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=24\\left(\\frac{d}{dθ}\\left(64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)\\right)+\\frac{d}{dθ}\\left(132\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)+\\frac{d}{dθ}\\left(14\\sec^{8}\\left(θ\\right)\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)\\right)=64\\left(4\\tan^{5}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+4\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=64\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the Product Rule: $$\\left(f{\\cdot}g\\right)'=f'{\\cdot}g+f{\\cdot}g'$$", "secondary": [ "$$f=\\sec^{4}\\left(θ\\right),\\:g=\\tan^{4}\\left(θ\\right)$$" ], "result": "=64\\left(\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)\\tan^{4}\\left(θ\\right)+\\frac{d}{dθ}\\left(\\tan^{4}\\left(θ\\right)\\right)\\sec^{4}\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Product%20Rule", "practiceTopic": "Product Rule" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)=4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{4},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{4}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{4}\\right)=4u^{3}$$", "input": "\\frac{d}{du}\\left(u^{4}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=4u^{4-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=4u^{3}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYl3oEe4EC6n4B8Xgz0cKnkqk3hxk9aCfAWodBRxXgUexuzo0Fl6VhneutEteidn+Kv8//6/nV5O4fb8Xgwi7mapNFxUvwBeni+JEIFAdbegzYEOcQgyW4n51cR7AeMAm7Q==" } }, { "type": "step", "result": "=4u^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgA2CO5cm49xNEqFV0DMaZJRqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidVduWEnoDYDg06IbaTsp9TFtqiuNhFTvmF8ZCDqWZNoubfnOah8gAvLMKOeNMctC5tVYZ/3r82LxPimKUFXjQFStvD8GKDNJJN8qiu0XRcVZAYAvbRKqkRVDaDBHkSBtDwYHkfmHHTze3JXk3LjiJp" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$4\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "4\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{3+1}\\left(θ\\right)$$" ], "result": "=4\\sec^{3+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$3+1=4$$", "result": "=4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7p9w76+/nvHitDrMIupHrrrPmAPxuc/RFHWXkB8A+mnqLeJKmeakO1+Q3aSIldG41PCyA7ixjCdfDYe+yac7KOvOIrkYdKW3Q58EnXaMU+7oslHPSvPf5UfEs5T+IFZ4XHjb2+5NLFZrsH9fcPWg/TXL71sHPAsAQtYZvjpaGSRwboK/Igc1IRC3EGruCoRzV1Zrc/SYRIR8JuxYpD+PJ+Q==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan^{4}\\left(θ\\right)\\right)=4\\tan^{3}\\left(θ\\right)\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan^{4}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}4\\left(\\tan\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan^{4}\\left(θ\\right)\\right)", "result": "=4\\left(\\tan\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{4},\\:\\:u=\\tan\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{4}\\right)\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{4}\\right)=4u^{3}$$", "input": "\\frac{d}{du}\\left(u^{4}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=4u^{4-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=4u^{3}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYl3oEe4EC6n4B8Xgz0cKnkqk3hxk9aCfAWodBRxXgUexuzo0Fl6VhneutEteidn+Kv8//6/nV5O4fb8Xgwi7mapNFxUvwBeni+JEIFAdbegzYEOcQgyW4n51cR7AeMAm7Q==" } }, { "type": "step", "result": "=4u^{3}\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\tan\\left(θ\\right)$$", "result": "=4\\left(\\tan\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYux7TmDJ3fopnHGBxoqO+pY2CO5cm49xNEqFV0DMaZJRqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidVduWEnoDYDg06IbaTsp9TzW39yg3f07n4Kh0WH31pB8q1KfCFKwSwA0qmWovUVP9tVYZ/3r82LxPimKUFXjQFStvD8GKDNJJN8qiu0XRcVZAYAvbRKqkRVDaDBHkSBtDwYHkfmHHTze3JXk3LjiJp" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "result": "=\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYs0Vsx3c0nt53Utu5DgHTe3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK00HV/bLwwglXz/2YNx2i4JHggSrmCoGjSdud4CyOvVOgdQN0k+N3NHNAkEIoLrhB631k80nn+ODwI4ID0kCF5+z7zRz81SqRD02HEUE4Wqh" } }, { "type": "step", "result": "=4\\left(\\tan\\left(θ\\right)\\right)^{3}\\sec^{2}\\left(θ\\right)" }, { "type": "step", "primary": "Simplify", "result": "=4\\tan^{3}\\left(θ\\right)\\sec^{2}\\left(θ\\right)", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=64\\left(4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{4}\\left(θ\\right)+4\\tan^{3}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Simplify $$64\\left(4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{4}\\left(θ\\right)+4\\tan^{3}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right):{\\quad}64\\left(4\\tan^{5}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+4\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)\\right)$$", "input": "64\\left(4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{4}\\left(θ\\right)+4\\tan^{3}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right)", "result": "=64\\left(4\\tan^{5}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+4\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "$$4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{4}\\left(θ\\right)=4\\tan^{5}\\left(θ\\right)\\sec^{4}\\left(θ\\right)$$", "input": "4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{4}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\tan\\left(θ\\right)\\tan^{4}\\left(θ\\right)=\\:\\tan^{1+4}\\left(θ\\right)$$" ], "result": "=4\\sec^{4}\\left(θ\\right)\\tan^{1+4}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+4=5$$", "result": "=4\\sec^{4}\\left(θ\\right)\\tan^{5}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7GHerUg4zB8o0EeBO6BKV162cmBc6Js8szDUhC2Hxdqhdbcltti6PzkTDSwkmBTY6q47vuWedXv2WUg94ER8IwVADhNIEz39BpXewdhMCBb/U7dZxYBZ6VQ59yTO/BTRekuHx4ulZqfxZHIsiQUpV1nyo9elyUFFpcPZW02cAt7Hms8g3dTKBs8Qcved1uTbvGp3//8xunxxjTjNOU2Uj8WAEODKzAB5RvbiyYeFhfmIkt3WiGR7ZaCaXvz77bMjS" } }, { "type": "interim", "title": "$$4\\tan^{3}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)=4\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)$$", "input": "4\\tan^{3}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)=\\:\\sec^{2+4}\\left(θ\\right)$$" ], "result": "=4\\tan^{3}\\left(θ\\right)\\sec^{2+4}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+4=6$$", "result": "=4\\tan^{3}\\left(θ\\right)\\sec^{6}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Q5Qt/9UZA+6+eHizBC+zZOH0qCTik4j9uS7jTF7ng2/8DCHl2/1cUFH2kQZvGLuVzRqDxPUzBN6vjj5oJL9kUH0lfPvVpT4vMUEVz1kntu7WxcLMxZwB9cGBLGbwbyjTrqr1t4QBx+t+Vf7xBGCGrcGZ/v+w0kROeeUMRN/Dzfvh9Kgk4pOI/bku40xe54NvceVwY//SmIC3EDK+C+Nj9Yx6/teVLREoXphSPTLZ2MaMe0O2wYArgGZd6UMSRoCP" } }, { "type": "step", "result": "=64\\left(4\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+4\\tan^{5}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7A5AfjU1SRWvT8rmMMzVgcBiMJmjxigW8LortZKvrqEZDn/Xi9tAk0BagOH0ipjvtP2VpgCWDQiLGOtegj6K/y/padcGORE5bHX37Io3viWolp/MB1OadflBmLQoq5Pmlo5FYteSPKwXny4uCMrdsK87yytEh92Rj5rzFYDrjNFetlKq927IFCE0mSgRkqm2pGWXl+3rXhjFFYJ9jApz9IbNYmw5UnCphczNkGo2TpwyBBTEk/JQ2cZ9WKuRzClU7VkZbu12YlUPSk9mOI6WLjyHQ+8WidKe5mLtnoi0iGZg+SAm6ZAjmyYLg+gLSGx4mXr7wSwGQCg6Xu6w7Iszv6TSei4QZxxrazp/cx7cYZvYA0hWLB/nt1m4ggrh+mxlbvzIPeEtDfcHv/z8uls8Teg==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(132\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)=132\\left(6\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+2\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(132\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=132\\frac{d}{dθ}\\left(\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the Product Rule: $$\\left(f{\\cdot}g\\right)'=f'{\\cdot}g+f{\\cdot}g'$$", "secondary": [ "$$f=\\sec^{6}\\left(θ\\right),\\:g=\\tan^{2}\\left(θ\\right)$$" ], "result": "=132\\left(\\frac{d}{dθ}\\left(\\sec^{6}\\left(θ\\right)\\right)\\tan^{2}\\left(θ\\right)+\\frac{d}{dθ}\\left(\\tan^{2}\\left(θ\\right)\\right)\\sec^{6}\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Product%20Rule", "practiceTopic": "Product Rule" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec^{6}\\left(θ\\right)\\right)=6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{6}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}6\\left(\\sec\\left(θ\\right)\\right)^{5}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{6}\\left(θ\\right)\\right)", "result": "=6\\left(\\sec\\left(θ\\right)\\right)^{5}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{6},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{6}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{6}\\right)=6u^{5}$$", "input": "\\frac{d}{du}\\left(u^{6}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=6u^{6-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=6u^{5}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYgOK76JOSNqGa1B1wU6ozWWk3hxk9aCfAWodBRxXgUexqSoDkuZ4Ab1St3bM8oqKcP8//6/nV5O4fb8Xgwi7mapNFxUvwBeni+JEIFAdbegzHfwjUTnSu4cWqw+8m79+NA==" } }, { "type": "step", "result": "=6u^{5}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=6\\left(\\sec\\left(θ\\right)\\right)^{5}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgBaJTGF5uPwSMzxoSyJNVZsqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKieMSe35zYqdvyntnEYxaQ88jJq7JZFtKgvxzzzuaqTGrebfnOah8gAvLMKOeNMctC5tVYZ/3r82LxPimKUFXjQFStvD8GKDNJJN8qiu0XRcVZAYAvbRKqkRVDaDBHkSBtDwYHkfmHHTze3JXk3LjiJp" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=6\\left(\\sec\\left(θ\\right)\\right)^{5}\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$6\\sec^{5}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "6\\sec^{5}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{5}\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{5+1}\\left(θ\\right)$$" ], "result": "=6\\sec^{5+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$5+1=6$$", "result": "=6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7QMJmgA2Sx2hYpGS5j/h1NLPmAPxuc/RFHWXkB8A+mnqLeJKmeakO1+Q3aSIldG417JiujyaxykLR7Tv0Nr+wOGprXgPFMmP3i5vY2oD+e4YslHPSvPf5UfEs5T+IFZ4XHjb2+5NLFZrsH9fcPWg/TRiIlnZT/wCWlhB0uSVNtvx/uR/EZqc6EbI3tnxSI5001Zrc/SYRIR8JuxYpD+PJ+Q==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan^{2}\\left(θ\\right)\\right)=2\\tan\\left(θ\\right)\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan^{2}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}2\\tan\\left(θ\\right)\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan^{2}\\left(θ\\right)\\right)", "result": "=2\\tan\\left(θ\\right)\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{2},\\:\\:u=\\tan\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{2}\\right)\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{2}\\right)=2u$$", "input": "\\frac{d}{du}\\left(u^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2u^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2u", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYr+VZUwBnLdzbS6DZQ+f4s+k3hxk9aCfAWodBRxXgUexMchyqTAJWrzJaDbnNcFsJUeCBKuYKgaNJ253gLI69U79qbCA2QqVmvm3jGRXZ2ppvbGT4j1utMEkCDH25m/vlQ==" } }, { "type": "step", "result": "=2u\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\tan\\left(θ\\right)$$", "result": "=2\\tan\\left(θ\\right)\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYux7TmDJ3fopnHGBxoqO+paSXIThcWH58oYoMPt18lzFqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidIn+HfeW3QQwAkRa15J0NSAlj501WLIv46IdbBaParYow2V0WdtbiweShXv1KOwqCjeh7+jKEzLb7VNCEMF3Z/Maswa5oIagKQFOfrnKm5+Gbb/PxX3awut5KxRxyZf8A=" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "result": "=\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYs0Vsx3c0nt53Utu5DgHTe3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK00HV/bLwwglXz/2YNx2i4JHggSrmCoGjSdud4CyOvVOgdQN0k+N3NHNAkEIoLrhB631k80nn+ODwI4ID0kCF5+z7zRz81SqRD02HEUE4Wqh" } }, { "type": "step", "result": "=2\\tan\\left(θ\\right)\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=132\\left(6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{2}\\left(θ\\right)+2\\tan\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{6}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Simplify $$132\\left(6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{2}\\left(θ\\right)+2\\tan\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{6}\\left(θ\\right)\\right):{\\quad}132\\left(6\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+2\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)\\right)$$", "input": "132\\left(6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{2}\\left(θ\\right)+2\\tan\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{6}\\left(θ\\right)\\right)", "result": "=132\\left(6\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+2\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "$$6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{2}\\left(θ\\right)=6\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)$$", "input": "6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\tan\\left(θ\\right)\\tan^{2}\\left(θ\\right)=\\:\\tan^{1+2}\\left(θ\\right)$$" ], "result": "=6\\sec^{6}\\left(θ\\right)\\tan^{1+2}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+2=3$$", "result": "=6\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7H4jwT8QhunkiDAGNb+Rcva2cmBc6Js8szDUhC2Hxdqi5HNbiQL8+p2P+VJTsJbU7q47vuWedXv2WUg94ER8IwZGREjArFZVzoA4Cy52193mbbDkw/o6UIkLjzolgTAsqwqP97mnHf8zpjilA0FhQHuj4b/xSLwXZQVzICbe8+7UtbJdlgVP3iSyKFg6v1zm6XXbfFRrmBUC90vtWO60GksdM6yd/xAZrmYAkuRY8/rYkt3WiGR7ZaCaXvz77bMjS" } }, { "type": "interim", "title": "$$2\\tan\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{6}\\left(θ\\right)=2\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "2\\tan\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{6}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{2}\\left(θ\\right)\\sec^{6}\\left(θ\\right)=\\:\\sec^{2+6}\\left(θ\\right)$$" ], "result": "=2\\tan\\left(θ\\right)\\sec^{2+6}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+6=8$$", "result": "=2\\tan\\left(θ\\right)\\sec^{8}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7OyotidIxGYYp2AND0rbhNaCGttAzkBWvgG18MKDLv3TbMUomGEDalcbJrsz1ZKu/q47vuWedXv2WUg94ER8IwSIYCdSNn6m2gNv8FkjgS1Wta2mu2dy0k1Wp64KXS/1rjT/+cTnIKvMqhzKaFBbxvzSei4QZxxrazp/cx7cYZvb1XqCxOosrrwiv/aS1ROCGSHdJEyJPexqhQEbR26e/s0wLhsnE4D1C6IGVVz/pqlY=" } }, { "type": "step", "result": "=132\\left(2\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)+6\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7PV+l92IXFa8mTDEwIfC7DCKI9Km/ZvzCnLF9eqmklKHqRJDJ/DeTzEayKJpmRlyWQi730E1b9JEo9956Qy3sPJgndhfIfsC9I7wDMrBBRj8AlilG71elit3w1IBbYN0PB5Bs5lVGb45zGNFiEvBODcjDyWbJQVk1pvf07Y462fJXV7Qqnb+eOc6LPNN3xs8nyZ0rztQs0QA5MTJACM+fadWc8BVevEVI1mgOPLFPH8CLGmNnLPWGf9PH3lpmjoJIZIRaf9fTURBjTQ6UkTGlImtPcLmP23q51gN6YJjM09cr8Xmq4XLWwD8V90BgYeZJP2VXWb3eVFnN+G/7GJIl8HXdshbMhQZCHpp35utrS0G0Y1gka5PyQUgt+7WzOOr4" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(14\\sec^{8}\\left(θ\\right)\\right)=112\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(14\\sec^{8}\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=14\\frac{d}{dθ}\\left(\\sec^{8}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Apply the chain rule:$${\\quad}8\\left(\\sec\\left(θ\\right)\\right)^{7}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{8}\\left(θ\\right)\\right)", "result": "=8\\left(\\sec\\left(θ\\right)\\right)^{7}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{8},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{8}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{8}\\right)=8u^{7}$$", "input": "\\frac{d}{du}\\left(u^{8}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=8u^{8-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=8u^{7}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqfCu3R3nHIhDvsPcLLz/1Kk3hxk9aCfAWodBRxXgUexxBsJp2NepDuy/SDgSGtTQP8//6/nV5O4fb8Xgwi7mapNFxUvwBeni+JEIFAdbegz9Cf5/xFJQkGUmQmSfOtWkA==" } }, { "type": "step", "result": "=8u^{7}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=8\\left(\\sec\\left(θ\\right)\\right)^{7}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgAmdV6hJ9cldZkSASENQvDKqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKie/FI4A8N6pVJJZl5fBw9jL2Bb9Z084y+BADz+fW/X+8ObfnOah8gAvLMKOeNMctC5tVYZ/3r82LxPimKUFXjQFStvD8GKDNJJN8qiu0XRcVZAYAvbRKqkRVDaDBHkSBtDwYHkfmHHTze3JXk3LjiJp" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=14\\cdot\\:8\\left(\\sec\\left(θ\\right)\\right)^{7}\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$14\\cdot\\:8\\sec^{7}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}112\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "14\\cdot\\:8\\sec^{7}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=112\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$14\\cdot\\:8=112$$", "result": "=112\\sec^{7}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{7}\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{7+1}\\left(θ\\right)$$" ], "result": "=112\\sec^{7+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$7+1=8$$", "result": "=112\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7qqpiC4le3DAJBVqCch5PKwVLcfLCp9Boesbd0nXQesBqKrdqZr3xi0oZ0kf4/81Y3XeO2tIUPH5Q2xrCOU6NXawGQ5eg9xcOzW/RgEfPor9DSTWuNGAj+wIwMGsd7LCW7lvKtANBUJdQPS8f9+853HKF3u2OIb4bFA3EO8aRlSVMbZv4Wa4I2LaPOrVXIZtj9A5Bkci853Hfrrw5WP/odGZiNlnz9jzyUG5gNa45hnA=" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=24\\left(64\\left(4\\tan^{5}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+4\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)\\right)+132\\left(6\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+2\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+112\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Expand $$64\\left(4\\tan^{5}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+4\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)\\right)+132\\left(6\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+2\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+112\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}256\\tan^{5}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+1048\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+376\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "64\\left(4\\tan^{5}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+4\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)\\right)+132\\left(6\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+2\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+112\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=24\\left(256\\tan^{5}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+1048\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+376\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Expand $$64\\left(4\\tan^{5}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+4\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)\\right):{\\quad}256\\tan^{5}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+256\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)$$", "input": "64\\left(4\\tan^{5}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+4\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)\\right)", "result": "=256\\tan^{5}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+256\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+132\\left(6\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+2\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+112\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b+c\\right)=ab+ac$$", "secondary": [ "$$a=64,\\:b=4\\tan^{5}\\left(θ\\right)\\sec^{4}\\left(θ\\right),\\:c=4\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)$$" ], "result": "=64\\cdot\\:4\\tan^{5}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+64\\cdot\\:4\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "primary": "Multiply the numbers: $$64\\cdot\\:4=256$$", "result": "=256\\tan^{5}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+256\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)" } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7YkvlWLayjKN6FXarLQ0GJcTHt7EfGTKANmJ5adlvOd46a6whJcHEYdFzx1pH94gcENAXTRYlkeXi/6iJJcYG3N6GQqufR6tr2vPxOUv7H+8zdEfU9seBjOhaIUijVXXbLyPYwflVgHzSz1oAHVz6pos4Mes11HpSwidsI6oRyK63hVfpOpUStEwbilPE4ogd+V7S6kFzr6zNGXs78LxaihJyf8zawtgEaDEKWrMLEzb8ak59hLGKbUsPp4qZlO8pI0RxNAAeCiAupyTRxbXafXxAop4KFNvJdALgGVyiwSDyPMNb+f0eisUGmCiaKoh74gBJl4WMO1rA0a30/bUYlg==" } }, { "type": "interim", "title": "Expand $$132\\left(6\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+2\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)\\right):{\\quad}792\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+264\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "132\\left(6\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+2\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)\\right)", "result": "=256\\tan^{5}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+256\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+792\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+264\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)+112\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b+c\\right)=ab+ac$$", "secondary": [ "$$a=132,\\:b=6\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right),\\:c=2\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)$$" ], "result": "=132\\cdot\\:6\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+132\\cdot\\:2\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "interim", "title": "Simplify $$132\\cdot\\:6\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+132\\cdot\\:2\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}792\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+264\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "132\\cdot\\:6\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+132\\cdot\\:2\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=792\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+264\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$132\\cdot\\:6=792$$", "result": "=792\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+132\\cdot\\:2\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "step", "primary": "Multiply the numbers: $$132\\cdot\\:2=264$$", "result": "=792\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+264\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7PV+l92IXFa8mTDEwIfC7DGQM6E4c9YPkyE7X/lieyos/6pCB86paeQ1o/mNgPa+jsFDSg/SgXmHLlF40pckVgc0ag8T1MwTer44+aCS/ZFArZHiZ7NuNPnvM2r6E9STNyZ0rztQs0QA5MTJACM+faQTXQmcLJlJVdaT/WTcsWRj8WaGp/4qrIjQsifAUsfjIo3oe/oyhMy2+1TQhDBd2f3FGtZiGdBdn/vMAKrUamk+2lifHQThNv1usau5svJJWtITS+u4dgApxM44qzMwAP0h3SRMiT3saoUBG0dunv7Ov8EPIWSoKCGyUa9XWGgj9" } }, { "type": "interim", "title": "Simplify $$256\\tan^{5}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+256\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+792\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+264\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)+112\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}256\\tan^{5}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+1048\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+376\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "256\\tan^{5}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+256\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+792\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+264\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)+112\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=256\\tan^{5}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+1048\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+376\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Add similar elements: $$256\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+792\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)=1048\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)$$", "result": "=256\\tan^{5}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+1048\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+264\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)+112\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "step", "primary": "Add similar elements: $$264\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)+112\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)=376\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=256\\tan^{5}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+1048\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+376\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=-8\\left(-4\\left(-4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+12\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)\\right)+24\\left(256\\tan^{5}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+1048\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+376\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=-8\\left(-4\\left(-4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+12\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)\\right)+24\\left(256\\tan^{5}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+1048\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+376\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "Evaluate $$-8\\left(-4\\left(-4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+12\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)\\right)+24\\left(256\\tan^{5}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+1048\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+376\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)\\right)\\:$$at point $$θ=0:{\\quad}0$$", "result": "=0", "steps": [ { "type": "step", "primary": "Take the point $$θ=0$$ and plug it into $$-8\\left(-4\\left(-4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+12\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)\\right)+24\\left(256\\tan^{5}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+1048\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+376\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)\\right)$$", "result": "=-8\\left(-4\\left(-4\\sec^{2}\\left(0\\right)\\tan\\left(0\\right)+12\\sec^{4}\\left(0\\right)\\tan\\left(0\\right)\\right)+12\\left(16\\sec^{4}\\left(0\\right)\\tan^{3}\\left(0\\right)+14\\sec^{6}\\left(0\\right)\\tan\\left(0\\right)\\right)\\right)+24\\left(256\\tan^{5}\\left(0\\right)\\sec^{4}\\left(0\\right)+1048\\sec^{6}\\left(0\\right)\\tan^{3}\\left(0\\right)+376\\sec^{8}\\left(0\\right)\\tan\\left(0\\right)\\right)" }, { "type": "interim", "title": "$$8\\left(-4\\left(-4\\sec^{2}\\left(0\\right)\\tan\\left(0\\right)+12\\sec^{4}\\left(0\\right)\\tan\\left(0\\right)\\right)+12\\left(16\\sec^{4}\\left(0\\right)\\tan^{3}\\left(0\\right)+14\\sec^{6}\\left(0\\right)\\tan\\left(0\\right)\\right)\\right)=0$$", "input": "8\\left(-4\\left(-4\\sec^{2}\\left(0\\right)\\tan\\left(0\\right)+12\\sec^{4}\\left(0\\right)\\tan\\left(0\\right)\\right)+12\\left(16\\sec^{4}\\left(0\\right)\\tan^{3}\\left(0\\right)+14\\sec^{6}\\left(0\\right)\\tan\\left(0\\right)\\right)\\right)", "steps": [ { "type": "interim", "title": "$$4\\left(-4\\sec^{2}\\left(0\\right)\\tan\\left(0\\right)+12\\sec^{4}\\left(0\\right)\\tan\\left(0\\right)\\right)=0$$", "input": "4\\left(-4\\sec^{2}\\left(0\\right)\\tan\\left(0\\right)+12\\sec^{4}\\left(0\\right)\\tan\\left(0\\right)\\right)", "steps": [ { "type": "interim", "title": "$$4\\sec^{2}\\left(0\\right)\\tan\\left(0\\right)=0$$", "input": "4\\sec^{2}\\left(0\\right)\\tan\\left(0\\right)", "steps": [ { "type": "interim", "title": "$$\\sec^{2}\\left(0\\right)=1$$", "input": "\\sec^{2}\\left(0\\right)", "steps": [ { "type": "interim", "title": "Simplify $$\\sec\\left(0\\right):{\\quad}1$$", "input": "\\sec\\left(0\\right)", "result": "=1^{2}", "steps": [ { "type": "step", "primary": "Use the following trivial identity:$${\\quad}\\sec\\left(0\\right)=1$$", "secondary": [ "$$\\sec\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sec(x)&x&\\sec(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{2\\sqrt{3}}{3}&\\frac{7π}{6}&-\\frac{2\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&\\sqrt{2}&\\frac{5π}{4}&-\\sqrt{2}\\\\\\hline \\frac{π}{3}&2&\\frac{4π}{3}&-2\\\\\\hline \\frac{π}{2}&\\mathrm{Undefined}&\\frac{3π}{2}&\\mathrm{Undefined}\\\\\\hline \\frac{2π}{3}&-2&\\frac{5π}{3}&2\\\\\\hline \\frac{3π}{4}&-\\sqrt{2}&\\frac{7π}{4}&\\sqrt{2}\\\\\\hline \\frac{5π}{6}&-\\frac{2\\sqrt{3}}{3}&\\frac{11π}{6}&\\frac{2\\sqrt{3}}{3}\\\\\\hline \\end{array}$$" ], "result": "=1" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7y1xaZeQTSzp609voOcYk6VXTSum/z5kLpMzXS1UJIexiYFOaxxQqg3o8CNKkTm2qW+Sc57RV820557NJqaC71BfVjTo90KMWspnEdE4CKjE=" } }, { "type": "step", "result": "=4\\cdot\\:1\\cdot\\:\\tan\\left(0\\right)" }, { "type": "interim", "title": "Simplify $$\\tan\\left(0\\right):{\\quad}0$$", "input": "\\tan\\left(0\\right)", "result": "=4\\cdot\\:1\\cdot\\:0", "steps": [ { "type": "step", "primary": "Use the following trivial identity:$${\\quad}\\tan\\left(0\\right)=0$$", "secondary": [ "$$\\tan\\left(x\\right)$$ periodicity table with $$πn$$ cycle:<br/>$$\\begin{array}{|c|c|}\\hline x&\\tan(x)\\\\\\hline 0&0\\\\\\hline \\frac{π}{6}&\\frac{\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&1\\\\\\hline \\frac{π}{3}&\\sqrt{3}\\\\\\hline \\frac{π}{2}&\\pm\\infty\\\\\\hline \\frac{2π}{3}&-\\sqrt{3}\\\\\\hline \\frac{3π}{4}&-1\\\\\\hline \\frac{5π}{6}&-\\frac{\\sqrt{3}}{3}\\\\\\hline &\\\\\\hline \\end{array}$$" ], "result": "=0" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Apply rule $$0\\cdot\\:a=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7yoGM/PYNKxD+4Xcq8hjhc6mPj3gXRowY53lqkfZS9MsJQJZuTAY5js+oqjdT8kslKXPrgUnq5rRq9Cvw1ceDci6Tg1fVs3FHX+2oeKw3d9PzoXLK02yt5K+3YXszYC4c" } }, { "type": "interim", "title": "$$12\\sec^{4}\\left(0\\right)\\tan\\left(0\\right)=0$$", "input": "12\\sec^{4}\\left(0\\right)\\tan\\left(0\\right)", "steps": [ { "type": "interim", "title": "$$\\sec^{4}\\left(0\\right)=1$$", "input": "\\sec^{4}\\left(0\\right)", "steps": [ { "type": "interim", "title": "Simplify $$\\sec\\left(0\\right):{\\quad}1$$", "input": "\\sec\\left(0\\right)", "result": "=1^{4}", "steps": [ { "type": "step", "primary": "Use the following trivial identity:$${\\quad}\\sec\\left(0\\right)=1$$", "secondary": [ "$$\\sec\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sec(x)&x&\\sec(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{2\\sqrt{3}}{3}&\\frac{7π}{6}&-\\frac{2\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&\\sqrt{2}&\\frac{5π}{4}&-\\sqrt{2}\\\\\\hline \\frac{π}{3}&2&\\frac{4π}{3}&-2\\\\\\hline \\frac{π}{2}&\\mathrm{Undefined}&\\frac{3π}{2}&\\mathrm{Undefined}\\\\\\hline \\frac{2π}{3}&-2&\\frac{5π}{3}&2\\\\\\hline \\frac{3π}{4}&-\\sqrt{2}&\\frac{7π}{4}&\\sqrt{2}\\\\\\hline \\frac{5π}{6}&-\\frac{2\\sqrt{3}}{3}&\\frac{11π}{6}&\\frac{2\\sqrt{3}}{3}\\\\\\hline \\end{array}$$" ], "result": "=1" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7TnbPBMsEd78kmmwfYDHqNVXTSum/z5kLpMzXS1UJIexiYFOaxxQqg3o8CNKkTm2qW+Sc57RV820557NJqaC71Ka6Ma3rg33eDIhfX/iuVnM=" } }, { "type": "step", "result": "=12\\cdot\\:1\\cdot\\:\\tan\\left(0\\right)" }, { "type": "interim", "title": "Simplify $$\\tan\\left(0\\right):{\\quad}0$$", "input": "\\tan\\left(0\\right)", "result": "=12\\cdot\\:1\\cdot\\:0", "steps": [ { "type": "step", "primary": "Use the following trivial identity:$${\\quad}\\tan\\left(0\\right)=0$$", "secondary": [ "$$\\tan\\left(x\\right)$$ periodicity table with $$πn$$ cycle:<br/>$$\\begin{array}{|c|c|}\\hline x&\\tan(x)\\\\\\hline 0&0\\\\\\hline \\frac{π}{6}&\\frac{\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&1\\\\\\hline \\frac{π}{3}&\\sqrt{3}\\\\\\hline \\frac{π}{2}&\\pm\\infty\\\\\\hline \\frac{2π}{3}&-\\sqrt{3}\\\\\\hline \\frac{3π}{4}&-1\\\\\\hline \\frac{5π}{6}&-\\frac{\\sqrt{3}}{3}\\\\\\hline &\\\\\\hline \\end{array}$$" ], "result": "=0" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Apply rule $$0\\cdot\\:a=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ztKQGqbHeZS/JvOABkqK/RjTuW+Q2J5Q5LUQGIymb9t1g99dC9fj9sg0EHzBIRDRd79UrkSVT0SCLs80Lgihl04wkUSqnBCRad8lKr40KATbzo7SL2lLjXVd+4Xv52R0JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=4\\left(0-0\\right)" }, { "type": "step", "primary": "Add/Subtract the numbers: $$-0+0=0$$", "result": "=4\\cdot\\:0" }, { "type": "step", "primary": "Apply rule $$0\\cdot\\:a=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s72ljAGTuL6Of2OboIiv8PbQnt1nd3Bhlv5dVInbaEGkWwKwfJEYKIuZZNasvHulbS9s43/aG4RqIhR8wn5cy6TATlqoY3+wtl96gVlHC0Yuo/y9DKGIPglJ+qMi9xDu2KugLLHAavdJagJJ1+8BBBh5aQ3x5UBSTkBtTLZJG1WFSzXF1PdZKGLQpjjNSl325EEEWJXNA8Rj59tQEICFmFoA==" } }, { "type": "interim", "title": "$$12\\left(16\\sec^{4}\\left(0\\right)\\tan^{3}\\left(0\\right)+14\\sec^{6}\\left(0\\right)\\tan\\left(0\\right)\\right)=0$$", "input": "12\\left(16\\sec^{4}\\left(0\\right)\\tan^{3}\\left(0\\right)+14\\sec^{6}\\left(0\\right)\\tan\\left(0\\right)\\right)", "steps": [ { "type": "interim", "title": "$$16\\sec^{4}\\left(0\\right)\\tan^{3}\\left(0\\right)=0$$", "input": "16\\sec^{4}\\left(0\\right)\\tan^{3}\\left(0\\right)", "steps": [ { "type": "interim", "title": "$$\\sec^{4}\\left(0\\right)=1$$", "input": "\\sec^{4}\\left(0\\right)", "steps": [ { "type": "interim", "title": "Simplify $$\\sec\\left(0\\right):{\\quad}1$$", "input": "\\sec\\left(0\\right)", "result": "=1^{4}", "steps": [ { "type": "step", "primary": "Use the following trivial identity:$${\\quad}\\sec\\left(0\\right)=1$$", "secondary": [ "$$\\sec\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sec(x)&x&\\sec(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{2\\sqrt{3}}{3}&\\frac{7π}{6}&-\\frac{2\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&\\sqrt{2}&\\frac{5π}{4}&-\\sqrt{2}\\\\\\hline \\frac{π}{3}&2&\\frac{4π}{3}&-2\\\\\\hline \\frac{π}{2}&\\mathrm{Undefined}&\\frac{3π}{2}&\\mathrm{Undefined}\\\\\\hline \\frac{2π}{3}&-2&\\frac{5π}{3}&2\\\\\\hline \\frac{3π}{4}&-\\sqrt{2}&\\frac{7π}{4}&\\sqrt{2}\\\\\\hline \\frac{5π}{6}&-\\frac{2\\sqrt{3}}{3}&\\frac{11π}{6}&\\frac{2\\sqrt{3}}{3}\\\\\\hline \\end{array}$$" ], "result": "=1" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7TnbPBMsEd78kmmwfYDHqNVXTSum/z5kLpMzXS1UJIexiYFOaxxQqg3o8CNKkTm2qW+Sc57RV820557NJqaC71Ka6Ma3rg33eDIhfX/iuVnM=" } }, { "type": "step", "result": "=16\\cdot\\:1\\cdot\\:\\tan^{3}\\left(0\\right)" }, { "type": "interim", "title": "$$\\tan^{3}\\left(0\\right)=0$$", "input": "\\tan^{3}\\left(0\\right)", "steps": [ { "type": "interim", "title": "Simplify $$\\tan\\left(0\\right):{\\quad}0$$", "input": "\\tan\\left(0\\right)", "result": "=0^{3}", "steps": [ { "type": "step", "primary": "Use the following trivial identity:$${\\quad}\\tan\\left(0\\right)=0$$", "secondary": [ "$$\\tan\\left(x\\right)$$ periodicity table with $$πn$$ cycle:<br/>$$\\begin{array}{|c|c|}\\hline x&\\tan(x)\\\\\\hline 0&0\\\\\\hline \\frac{π}{6}&\\frac{\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&1\\\\\\hline \\frac{π}{3}&\\sqrt{3}\\\\\\hline \\frac{π}{2}&\\pm\\infty\\\\\\hline \\frac{2π}{3}&-\\sqrt{3}\\\\\\hline \\frac{3π}{4}&-1\\\\\\hline \\frac{5π}{6}&-\\frac{\\sqrt{3}}{3}\\\\\\hline &\\\\\\hline \\end{array}$$" ], "result": "=0" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Apply rule $$0^{a}=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7GE9JoQhnOmDIpJSiT8FGMFXTSum/z5kLpMzXS1UJIeyqjfZqhkrOoAz8sRb7wXZWPGHkI0MFUiTEwg/g9Re5WAQEEAYC9h/Nb4xd/kIOxRQ=" } }, { "type": "step", "result": "=16\\cdot\\:1\\cdot\\:0" }, { "type": "step", "primary": "Apply rule $$0\\cdot\\:a=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7AhXsSCpYfSFLoEvOM6cQkr/Ehopp9rGqiIQLEoitJ5IgJ/ZZA32ZInFBpDtxBfiKf7LqB9CcyvYCWDsGseX09k29cjXxPd/F95KFhHS4hjheH6PVg9CLFp1ktSOp93NiCFX7kPfEkIuYfqIFW07ybg==" } }, { "type": "interim", "title": "$$14\\sec^{6}\\left(0\\right)\\tan\\left(0\\right)=0$$", "input": "14\\sec^{6}\\left(0\\right)\\tan\\left(0\\right)", "steps": [ { "type": "interim", "title": "$$\\sec^{6}\\left(0\\right)=1$$", "input": "\\sec^{6}\\left(0\\right)", "steps": [ { "type": "interim", "title": "Simplify $$\\sec\\left(0\\right):{\\quad}1$$", "input": "\\sec\\left(0\\right)", "result": "=1^{6}", "steps": [ { "type": "step", "primary": "Use the following trivial identity:$${\\quad}\\sec\\left(0\\right)=1$$", "secondary": [ "$$\\sec\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sec(x)&x&\\sec(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{2\\sqrt{3}}{3}&\\frac{7π}{6}&-\\frac{2\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&\\sqrt{2}&\\frac{5π}{4}&-\\sqrt{2}\\\\\\hline \\frac{π}{3}&2&\\frac{4π}{3}&-2\\\\\\hline \\frac{π}{2}&\\mathrm{Undefined}&\\frac{3π}{2}&\\mathrm{Undefined}\\\\\\hline \\frac{2π}{3}&-2&\\frac{5π}{3}&2\\\\\\hline \\frac{3π}{4}&-\\sqrt{2}&\\frac{7π}{4}&\\sqrt{2}\\\\\\hline \\frac{5π}{6}&-\\frac{2\\sqrt{3}}{3}&\\frac{11π}{6}&\\frac{2\\sqrt{3}}{3}\\\\\\hline \\end{array}$$" ], "result": "=1" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7WcSYqbVPn/qsbXXyqHdMbFXTSum/z5kLpMzXS1UJIexiYFOaxxQqg3o8CNKkTm2qW+Sc57RV820557NJqaC71Idwe88IvH72vjqcga7gyBg=" } }, { "type": "step", "result": "=14\\cdot\\:1\\cdot\\:\\tan\\left(0\\right)" }, { "type": "interim", "title": "Simplify $$\\tan\\left(0\\right):{\\quad}0$$", "input": "\\tan\\left(0\\right)", "result": "=14\\cdot\\:1\\cdot\\:0", "steps": [ { "type": "step", "primary": "Use the following trivial identity:$${\\quad}\\tan\\left(0\\right)=0$$", "secondary": [ "$$\\tan\\left(x\\right)$$ periodicity table with $$πn$$ cycle:<br/>$$\\begin{array}{|c|c|}\\hline x&\\tan(x)\\\\\\hline 0&0\\\\\\hline \\frac{π}{6}&\\frac{\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&1\\\\\\hline \\frac{π}{3}&\\sqrt{3}\\\\\\hline \\frac{π}{2}&\\pm\\infty\\\\\\hline \\frac{2π}{3}&-\\sqrt{3}\\\\\\hline \\frac{3π}{4}&-1\\\\\\hline \\frac{5π}{6}&-\\frac{\\sqrt{3}}{3}\\\\\\hline &\\\\\\hline \\end{array}$$" ], "result": "=0" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Apply rule $$0\\cdot\\:a=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7uESehsDSxkrMlWGVYh0nEhjTuW+Q2J5Q5LUQGIymb9t1g99dC9fj9sg0EHzBIRDRd79UrkSVT0SCLs80Lgihl/MHZz/q3kUqOBwvqGYpXX/bzo7SL2lLjXVd+4Xv52R0JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=12\\left(0+0\\right)" }, { "type": "step", "primary": "Add the numbers: $$0+0=0$$", "result": "=12\\cdot\\:0" }, { "type": "step", "primary": "Apply rule $$0\\cdot\\:a=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7THpUEnejLq8ujSxRrFRF77vrZMykvfhAOYSGWz3LDed1Xm1xDLBN/I9hH5mwedz1q3Z1IfyMnAKE4AfR1MbC4glAlm5MBjmOz6iqN1PySyUpc+uBSermtGr0K/DVx4Ny6jvDJvcRv402CzqvSw1xBNxzzDZrUHthRg6PFIysvL6g1+/Xb7zhBvdvyquoZxFY+mnfm1+QDeBe9hfT/VE9PA==" } }, { "type": "step", "result": "=8\\left(0-0\\right)" }, { "type": "step", "primary": "Add/Subtract the numbers: $$-0+0=0$$", "result": "=8\\cdot\\:0" }, { "type": "step", "primary": "Apply rule $$0\\cdot\\:a=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7EbWaeCR2s6GCwM26QHJgh+MtZk1apgXS2PnMY00HuHozq+QS9kblohPScFnBjh/0Fo5ZU2babDpl+ccq2CFCxNsIZOXGNZjtfrh0bAM8BJq+2an+8KoqGiBENhibzAnjuBkju6iNwU4VC61166Tx7CAn9lkDfZkicUGkO3EF+Ip/suoH0JzK9gJYOwax5fT264kbgUE1y9zJdMNawi/5G8XaLmKEwIMKuUdLlLEit0PoVLYy6zSMwewZXjTUiS9Y6dOLLCSRaLTut1U9XBz0lLyR1AGcpHddPfF3UQGRbAPX1j21YoaKCyN+9P+SWOjj0KIqH/3G71QYFJLrLnLwDfjla9rmOavR/vD7yUItxvA=" } }, { "type": "interim", "title": "$$24\\left(256\\tan^{5}\\left(0\\right)\\sec^{4}\\left(0\\right)+1048\\sec^{6}\\left(0\\right)\\tan^{3}\\left(0\\right)+376\\sec^{8}\\left(0\\right)\\tan\\left(0\\right)\\right)=0$$", "input": "24\\left(256\\tan^{5}\\left(0\\right)\\sec^{4}\\left(0\\right)+1048\\sec^{6}\\left(0\\right)\\tan^{3}\\left(0\\right)+376\\sec^{8}\\left(0\\right)\\tan\\left(0\\right)\\right)", "steps": [ { "type": "interim", "title": "$$256\\tan^{5}\\left(0\\right)\\sec^{4}\\left(0\\right)=0$$", "input": "256\\tan^{5}\\left(0\\right)\\sec^{4}\\left(0\\right)", "steps": [ { "type": "interim", "title": "$$\\tan^{5}\\left(0\\right)=0$$", "input": "\\tan^{5}\\left(0\\right)", "steps": [ { "type": "interim", "title": "Simplify $$\\tan\\left(0\\right):{\\quad}0$$", "input": "\\tan\\left(0\\right)", "result": "=0^{5}", "steps": [ { "type": "step", "primary": "Use the following trivial identity:$${\\quad}\\tan\\left(0\\right)=0$$", "secondary": [ "$$\\tan\\left(x\\right)$$ periodicity table with $$πn$$ cycle:<br/>$$\\begin{array}{|c|c|}\\hline x&\\tan(x)\\\\\\hline 0&0\\\\\\hline \\frac{π}{6}&\\frac{\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&1\\\\\\hline \\frac{π}{3}&\\sqrt{3}\\\\\\hline \\frac{π}{2}&\\pm\\infty\\\\\\hline \\frac{2π}{3}&-\\sqrt{3}\\\\\\hline \\frac{3π}{4}&-1\\\\\\hline \\frac{5π}{6}&-\\frac{\\sqrt{3}}{3}\\\\\\hline &\\\\\\hline \\end{array}$$" ], "result": "=0" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Apply rule $$0^{a}=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7F1bxsbpxvPLF+iyLoerjQVXTSum/z5kLpMzXS1UJIeyqjfZqhkrOoAz8sRb7wXZWPGHkI0MFUiTEwg/g9Re5WE4REHkd4oo/aW+k17Q9Yes=" } }, { "type": "step", "result": "=256\\cdot\\:0\\cdot\\:\\sec^{4}\\left(0\\right)" }, { "type": "interim", "title": "$$\\sec^{4}\\left(0\\right)=1$$", "input": "\\sec^{4}\\left(0\\right)", "steps": [ { "type": "interim", "title": "Simplify $$\\sec\\left(0\\right):{\\quad}1$$", "input": "\\sec\\left(0\\right)", "result": "=1^{4}", "steps": [ { "type": "step", "primary": "Use the following trivial identity:$${\\quad}\\sec\\left(0\\right)=1$$", "secondary": [ "$$\\sec\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sec(x)&x&\\sec(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{2\\sqrt{3}}{3}&\\frac{7π}{6}&-\\frac{2\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&\\sqrt{2}&\\frac{5π}{4}&-\\sqrt{2}\\\\\\hline \\frac{π}{3}&2&\\frac{4π}{3}&-2\\\\\\hline \\frac{π}{2}&\\mathrm{Undefined}&\\frac{3π}{2}&\\mathrm{Undefined}\\\\\\hline \\frac{2π}{3}&-2&\\frac{5π}{3}&2\\\\\\hline \\frac{3π}{4}&-\\sqrt{2}&\\frac{7π}{4}&\\sqrt{2}\\\\\\hline \\frac{5π}{6}&-\\frac{2\\sqrt{3}}{3}&\\frac{11π}{6}&\\frac{2\\sqrt{3}}{3}\\\\\\hline \\end{array}$$" ], "result": "=1" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7TnbPBMsEd78kmmwfYDHqNVXTSum/z5kLpMzXS1UJIexiYFOaxxQqg3o8CNKkTm2qW+Sc57RV820557NJqaC71Ka6Ma3rg33eDIhfX/iuVnM=" } }, { "type": "step", "result": "=256\\cdot\\:0\\cdot\\:1" }, { "type": "step", "primary": "Apply rule $$0\\cdot\\:a=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7yISWgNr97nahrHDSfrYHCYcqLPt0wQ3uXcT07CJiPw7ehkKrn0era9rz8TlL+x/vZuJKdCFsPJy1+5gBMEc9dkeSOCNGjtYTYUbvIbVyNKeCRDloPPHMk6g3xYXOmDbRh1+wup2q1UW0KWBCS2BVHw==" } }, { "type": "interim", "title": "$$1048\\sec^{6}\\left(0\\right)\\tan^{3}\\left(0\\right)=0$$", "input": "1048\\sec^{6}\\left(0\\right)\\tan^{3}\\left(0\\right)", "steps": [ { "type": "interim", "title": "$$\\sec^{6}\\left(0\\right)=1$$", "input": "\\sec^{6}\\left(0\\right)", "steps": [ { "type": "interim", "title": "Simplify $$\\sec\\left(0\\right):{\\quad}1$$", "input": "\\sec\\left(0\\right)", "result": "=1^{6}", "steps": [ { "type": "step", "primary": "Use the following trivial identity:$${\\quad}\\sec\\left(0\\right)=1$$", "secondary": [ "$$\\sec\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sec(x)&x&\\sec(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{2\\sqrt{3}}{3}&\\frac{7π}{6}&-\\frac{2\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&\\sqrt{2}&\\frac{5π}{4}&-\\sqrt{2}\\\\\\hline \\frac{π}{3}&2&\\frac{4π}{3}&-2\\\\\\hline \\frac{π}{2}&\\mathrm{Undefined}&\\frac{3π}{2}&\\mathrm{Undefined}\\\\\\hline \\frac{2π}{3}&-2&\\frac{5π}{3}&2\\\\\\hline \\frac{3π}{4}&-\\sqrt{2}&\\frac{7π}{4}&\\sqrt{2}\\\\\\hline \\frac{5π}{6}&-\\frac{2\\sqrt{3}}{3}&\\frac{11π}{6}&\\frac{2\\sqrt{3}}{3}\\\\\\hline \\end{array}$$" ], "result": "=1" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7WcSYqbVPn/qsbXXyqHdMbFXTSum/z5kLpMzXS1UJIexiYFOaxxQqg3o8CNKkTm2qW+Sc57RV820557NJqaC71Idwe88IvH72vjqcga7gyBg=" } }, { "type": "step", "result": "=1048\\cdot\\:1\\cdot\\:\\tan^{3}\\left(0\\right)" }, { "type": "interim", "title": "$$\\tan^{3}\\left(0\\right)=0$$", "input": "\\tan^{3}\\left(0\\right)", "steps": [ { "type": "interim", "title": "Simplify $$\\tan\\left(0\\right):{\\quad}0$$", "input": "\\tan\\left(0\\right)", "result": "=0^{3}", "steps": [ { "type": "step", "primary": "Use the following trivial identity:$${\\quad}\\tan\\left(0\\right)=0$$", "secondary": [ "$$\\tan\\left(x\\right)$$ periodicity table with $$πn$$ cycle:<br/>$$\\begin{array}{|c|c|}\\hline x&\\tan(x)\\\\\\hline 0&0\\\\\\hline \\frac{π}{6}&\\frac{\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&1\\\\\\hline \\frac{π}{3}&\\sqrt{3}\\\\\\hline \\frac{π}{2}&\\pm\\infty\\\\\\hline \\frac{2π}{3}&-\\sqrt{3}\\\\\\hline \\frac{3π}{4}&-1\\\\\\hline \\frac{5π}{6}&-\\frac{\\sqrt{3}}{3}\\\\\\hline &\\\\\\hline \\end{array}$$" ], "result": "=0" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Apply rule $$0^{a}=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7GE9JoQhnOmDIpJSiT8FGMFXTSum/z5kLpMzXS1UJIeyqjfZqhkrOoAz8sRb7wXZWPGHkI0MFUiTEwg/g9Re5WAQEEAYC9h/Nb4xd/kIOxRQ=" } }, { "type": "step", "result": "=1048\\cdot\\:1\\cdot\\:0" }, { "type": "step", "primary": "Apply rule $$0\\cdot\\:a=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7P4Tm2SVHynj/82XBtL3J9KXddIUciaOqdOOpGhjvd/dV00rpv8+ZC6TM10tVCSHsqo32aoZKzqAM/LEW+8F2VmvYkaJ+syIsVymr0zTkUaRZXo5elHJQ7S0xAMPEX14gBAQQBgL2H81vjF3+Qg7FFA==" } }, { "type": "interim", "title": "$$376\\sec^{8}\\left(0\\right)\\tan\\left(0\\right)=0$$", "input": "376\\sec^{8}\\left(0\\right)\\tan\\left(0\\right)", "steps": [ { "type": "interim", "title": "$$\\sec^{8}\\left(0\\right)=1$$", "input": "\\sec^{8}\\left(0\\right)", "steps": [ { "type": "interim", "title": "Simplify $$\\sec\\left(0\\right):{\\quad}1$$", "input": "\\sec\\left(0\\right)", "result": "=1^{8}", "steps": [ { "type": "step", "primary": "Use the following trivial identity:$${\\quad}\\sec\\left(0\\right)=1$$", "secondary": [ "$$\\sec\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sec(x)&x&\\sec(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{2\\sqrt{3}}{3}&\\frac{7π}{6}&-\\frac{2\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&\\sqrt{2}&\\frac{5π}{4}&-\\sqrt{2}\\\\\\hline \\frac{π}{3}&2&\\frac{4π}{3}&-2\\\\\\hline \\frac{π}{2}&\\mathrm{Undefined}&\\frac{3π}{2}&\\mathrm{Undefined}\\\\\\hline \\frac{2π}{3}&-2&\\frac{5π}{3}&2\\\\\\hline \\frac{3π}{4}&-\\sqrt{2}&\\frac{7π}{4}&\\sqrt{2}\\\\\\hline \\frac{5π}{6}&-\\frac{2\\sqrt{3}}{3}&\\frac{11π}{6}&\\frac{2\\sqrt{3}}{3}\\\\\\hline \\end{array}$$" ], "result": "=1" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7p6IxEfKejassVlEaaYgXqFXTSum/z5kLpMzXS1UJIexiYFOaxxQqg3o8CNKkTm2qW+Sc57RV820557NJqaC71Dd1z7LYeV9QGyBQ1bCNf14=" } }, { "type": "step", "result": "=376\\cdot\\:1\\cdot\\:\\tan\\left(0\\right)" }, { "type": "interim", "title": "Simplify $$\\tan\\left(0\\right):{\\quad}0$$", "input": "\\tan\\left(0\\right)", "result": "=376\\cdot\\:1\\cdot\\:0", "steps": [ { "type": "step", "primary": "Use the following trivial identity:$${\\quad}\\tan\\left(0\\right)=0$$", "secondary": [ "$$\\tan\\left(x\\right)$$ periodicity table with $$πn$$ cycle:<br/>$$\\begin{array}{|c|c|}\\hline x&\\tan(x)\\\\\\hline 0&0\\\\\\hline \\frac{π}{6}&\\frac{\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&1\\\\\\hline \\frac{π}{3}&\\sqrt{3}\\\\\\hline \\frac{π}{2}&\\pm\\infty\\\\\\hline \\frac{2π}{3}&-\\sqrt{3}\\\\\\hline \\frac{3π}{4}&-1\\\\\\hline \\frac{5π}{6}&-\\frac{\\sqrt{3}}{3}\\\\\\hline &\\\\\\hline \\end{array}$$" ], "result": "=0" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Apply rule $$0\\cdot\\:a=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7F+xD+aB6xTPieXwact4KRP+hRajxLBthlPAhEWP8QfcDnzlbPZjyKgy1eUCFsLd5uA0vbJw0XRhSBCaIjQqkFimSgKqnsKNpQFRZsFRFMDM1KQiy23quw6LZdnjMrNLGsIjaxJ4DvjTb2fbKjbvtlQ==" } }, { "type": "step", "result": "=24\\left(0+0+0\\right)" }, { "type": "step", "primary": "Add the numbers: $$0+0+0=0$$", "result": "=24\\cdot\\:0" }, { "type": "step", "primary": "Apply rule $$0\\cdot\\:a=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7BfYBhg8Jjf8r0LyT98P/Dgf3ZIV87m0hl5+kJdelMZCdkh5mjAOvA2zsKqvBLEreFOU1VzBhsv4pKjsQMe9XQ7O3oT6e2Es27cv+MzTwuNk9UVXzstM1oSzRK26cGrhLq47vuWedXv2WUg94ER8IwaYp+18uqAXrMTpLgOZPq6TAjbKETcL5MvyG3mc/E3nZZJZ0ekoUL3bOXqK7cfFAaqLBniLVwBHurOcM+5P4bXq762TMpL34QDmEhls9yw3n0HELEymiBL2jPLbalswE1lld0gqt5LgxI4dRnJHsnuE=" } }, { "type": "step", "result": "=-0+0" }, { "type": "step", "primary": "Simplify" }, { "type": "step", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Laplace Eval Function At Point Specific 3Eq" } }, { "type": "step", "result": "0" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\frac{d^{9}}{dθ^{9}}\\left(\\tan\\left(θ\\right)\\right)\\left(0\\right){\\quad:\\quad}7936$$", "input": "\\frac{d^{9}}{dθ^{9}}\\left(\\tan\\left(θ\\right)\\right)\\left(0\\right)", "steps": [ { "type": "interim", "title": "$$\\frac{d^{9}}{dθ^{9}}\\left(\\tan\\left(θ\\right)\\right)=-8\\left(-4\\left(-4\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)+12\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)\\right)+12\\left(64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+132\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+14\\sec^{8}\\left(θ\\right)\\right)\\right)+24\\left(7568\\sec^{6}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+1024\\tan^{6}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+6152\\sec^{8}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+376\\sec^{10}\\left(θ\\right)\\right)$$", "input": "\\frac{d^{9}}{dθ^{9}}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "result": "=\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYs0Vsx3c0nt53Utu5DgHTe3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK00HV/bLwwglXz/2YNx2i4JHggSrmCoGjSdud4CyOvVOgdQN0k+N3NHNAkEIoLrhB631k80nn+ODwI4ID0kCF5+z7zRz81SqRD02HEUE4Wqh" } }, { "type": "step", "result": "=\\frac{d^{8}}{dθ^{8}}\\left(\\sec^{2}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{2},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{2}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{2}\\right)=2u$$", "input": "\\frac{d}{du}\\left(u^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2u^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2u", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYr+VZUwBnLdzbS6DZQ+f4s+k3hxk9aCfAWodBRxXgUexMchyqTAJWrzJaDbnNcFsJUeCBKuYKgaNJ253gLI69U79qbCA2QqVmvm3jGRXZ2ppvbGT4j1utMEkCDH25m/vlQ==" } }, { "type": "step", "result": "=2u\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgCSXIThcWH58oYoMPt18lzFqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidZClirlWNFPnHR1rvdyoFLI7IC+oqnwb1T6nPLpiw1eysw2NXPd8jZnr3fcPpqc7ejeh7+jKEzLb7VNCEMF3Z/Maswa5oIagKQFOfrnKm5+Gbb/PxX3awut5KxRxyZf8A=" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{1+1}\\left(θ\\right)$$" ], "result": "=2\\sec^{1+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7aY4rYekMZhUfP6sZ6mfqd6Lsz0lt1XLEr88Kc/cjJo5V00rpv8+ZC6TM10tVCSHspqrCcTwb/I0f8OwMjuU6eBAUa+AHsHd5OGpMiuFITjOuqvW3hAHH635V/vEEYIatHimBRYRqHSWeJkuUPhfTC5LQ0B1JNgsb5SXzbS+SCnAfGB9H25Zc+Ua1z7Kw8kgb4gBJl4WMO1rA0a30/bUYlg==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{d^{7}}{dθ^{7}}\\left(2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\right)=-4\\sec^{2}\\left(θ\\right)+6\\sec^{4}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=2\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the Product Rule: $$\\left(f{\\cdot}g\\right)'=f'{\\cdot}g+f{\\cdot}g'$$", "secondary": [ "$$f=\\sec^{2}\\left(θ\\right),\\:g=\\tan\\left(θ\\right)$$" ], "result": "=2\\left(\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)\\tan\\left(θ\\right)+\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)\\sec^{2}\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Product%20Rule", "practiceTopic": "Product Rule" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{2},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{2}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{2}\\right)=2u$$", "input": "\\frac{d}{du}\\left(u^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2u^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2u", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYr+VZUwBnLdzbS6DZQ+f4s+k3hxk9aCfAWodBRxXgUexMchyqTAJWrzJaDbnNcFsJUeCBKuYKgaNJ253gLI69U79qbCA2QqVmvm3jGRXZ2ppvbGT4j1utMEkCDH25m/vlQ==" } }, { "type": "step", "result": "=2u\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgCSXIThcWH58oYoMPt18lzFqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidZClirlWNFPnHR1rvdyoFLI7IC+oqnwb1T6nPLpiw1eysw2NXPd8jZnr3fcPpqc7ejeh7+jKEzLb7VNCEMF3Z/Maswa5oIagKQFOfrnKm5+Gbb/PxX3awut5KxRxyZf8A=" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{1+1}\\left(θ\\right)$$" ], "result": "=2\\sec^{1+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7aY4rYekMZhUfP6sZ6mfqd6Lsz0lt1XLEr88Kc/cjJo5V00rpv8+ZC6TM10tVCSHspqrCcTwb/I0f8OwMjuU6eBAUa+AHsHd5OGpMiuFITjOuqvW3hAHH635V/vEEYIatHimBRYRqHSWeJkuUPhfTC5LQ0B1JNgsb5SXzbS+SCnAfGB9H25Zc+Ua1z7Kw8kgb4gBJl4WMO1rA0a30/bUYlg==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "result": "=\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYs0Vsx3c0nt53Utu5DgHTe3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK00HV/bLwwglXz/2YNx2i4JHggSrmCoGjSdud4CyOvVOgdQN0k+N3NHNAkEIoLrhB631k80nn+ODwI4ID0kCF5+z7zRz81SqRD02HEUE4Wqh" } }, { "type": "step", "result": "=2\\left(2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Simplify $$2\\left(2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\right):{\\quad}-4\\sec^{2}\\left(θ\\right)+6\\sec^{4}\\left(θ\\right)$$", "input": "2\\left(2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\right)", "result": "=-4\\sec^{2}\\left(θ\\right)+6\\sec^{4}\\left(θ\\right)", "steps": [ { "type": "interim", "title": "$$2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)=2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)$$", "input": "2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\tan\\left(θ\\right)\\tan\\left(θ\\right)=\\:\\tan^{1+1}\\left(θ\\right)$$" ], "result": "=2\\sec^{2}\\left(θ\\right)\\tan^{1+1}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Md1SJKHYLxhOYRRbsLTu+axUZgt6FTmFSHnRsotkqemLeJKmeakO1+Q3aSIldG41s7BwSVdKEYSqn+4fKAQoGk0xmFIl1KTO5aAkhaHvKzBALhShbMGZIZsZWTO721HsWI4dG63F/aSz/PFDvu5FnO3sngYNzcydIedGQjjtM8sBMUhkLMqDD6KHvgZAgKMdRQzIIgXxnWeDhDTk2Lfum8RPUSGriaFRn0NZXHZrPQw=" } }, { "type": "interim", "title": "$$\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)=\\sec^{4}\\left(θ\\right)$$", "input": "\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)=\\:\\sec^{2+2}\\left(θ\\right)$$" ], "result": "=\\sec^{2+2}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+2=4$$", "result": "=\\sec^{4}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7AtrsReczPdUqIdn5/Ev5mFtNfVJqszzpMwg6spHO8F8gJ/ZZA32ZInFBpDtxBfiKn3Jyj2VJgBojKQHjINpkkETYvnjqwhFu6aWmDOYpAVMO/U6qyRZonn1WPQ/21pz3cKXVLLQXMot2gtnZH7qY5zkbLioo7YVO0SRnDAO/7Uokt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "=2\\left(\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Expand $$\\left(\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)\\cdot\\:2:{\\quad}2\\sec^{4}\\left(θ\\right)+4\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)$$", "input": "\\left(\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)\\cdot\\:2", "result": "=2\\sec^{4}\\left(θ\\right)+4\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)", "steps": [ { "type": "step", "result": "=2\\left(\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the distributive law: $$a\\left(b+c\\right)=ab+ac$$", "secondary": [ "$$a=2,\\:b=\\sec^{4}\\left(θ\\right),\\:c=2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)$$" ], "result": "=2\\sec^{4}\\left(θ\\right)+2\\cdot\\:2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=2\\sec^{4}\\left(θ\\right)+4\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7cTiUO1fOFax2wXHZewjhxEUAdh/O2PLAxzMjSpKce9YL4tWjzbYCZhBh/Hc2rUXacMKakGXZAry+N/hvJOodjXCQoYlYQ8U+Tfyx0kyzI8iF3F2TkMs7mKjIsUPAR/Vd82dH/qvnCC7/oZy4HZlX2aHh6acbPp8G70jj/skoFx7vbBmbuQNTF0TphKZ8Ruvacu+LCkBvFQsw7xBL2Ys2TUYb1899n2cPVSJTSzzdZzp+38s+uQz1W6FLDk3KkIYWhJuRFkG6Ai089lDjeB1lMbCI2sSeA74029n2yo277ZU=" } }, { "type": "interim", "title": "Rewrite using trig identities", "input": "2\\sec^{4}\\left(θ\\right)+4\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)", "result": "=-4\\sec^{2}\\left(θ\\right)+6\\sec^{4}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Use the Pythagorean identity: $$\\tan^{2}\\left(x\\right)+1=\\sec^{2}\\left(x\\right)$$", "secondary": [ "$$\\tan^{2}\\left(x\\right)=\\sec^{2}\\left(x\\right)-1$$" ], "result": "=2\\sec^{4}\\left(θ\\right)+4\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right)" }, { "type": "interim", "title": "Simplify $$2\\sec^{4}\\left(θ\\right)+4\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right):{\\quad}6\\sec^{4}\\left(θ\\right)-4\\sec^{2}\\left(θ\\right)$$", "input": "2\\sec^{4}\\left(θ\\right)+4\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right)", "result": "=6\\sec^{4}\\left(θ\\right)-4\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "interim", "title": "Expand $$4\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right):{\\quad}4\\sec^{4}\\left(θ\\right)-4\\sec^{2}\\left(θ\\right)$$", "input": "4\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right)", "result": "=2\\sec^{4}\\left(θ\\right)+4\\sec^{4}\\left(θ\\right)-4\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b-c\\right)=ab-ac$$", "secondary": [ "$$a=4\\sec^{2}\\left(θ\\right),\\:b=\\sec^{2}\\left(θ\\right),\\:c=1$$" ], "result": "=4\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)-4\\sec^{2}\\left(θ\\right)\\cdot\\:1", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "result": "=4\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)-4\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$4\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)-4\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right):{\\quad}4\\sec^{4}\\left(θ\\right)-4\\sec^{2}\\left(θ\\right)$$", "input": "4\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)-4\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right)", "result": "=4\\sec^{4}\\left(θ\\right)-4\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "interim", "title": "$$4\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)=4\\sec^{4}\\left(θ\\right)$$", "input": "4\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)=\\:\\sec^{2+2}\\left(θ\\right)$$" ], "result": "=4\\sec^{2+2}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+2=4$$", "result": "=4\\sec^{4}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vjDr4p6TvcgiSLV1uDo4U9JYPDPDesonRI520G8zYRPehkKrn0era9rz8TlL+x/vMHVZf8xiryCPkbUIYXhiIHRfFgXB+vlRb/AHCJklpF9e7YF3TFAJSjcLX2VvhbubqX/cSuhK5Ty2xqd/IdNr6zlZ8CVoKZl2BAoDdAtcFNxU2fg2Eo8khoQfnK/P3ock" } }, { "type": "interim", "title": "$$4\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right)=4\\sec^{2}\\left(θ\\right)$$", "input": "4\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:1=4$$", "result": "=4\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7xF9rlQkuAOrRzKTf6eQs7fgkIqVBCzsTrV5i/ECNCw8tOtZYwUjyXhDTsNnn6Elr/BCsllMMEyc8UGKoUwuYxu6vfsn/ogOG+fHMfNQh9JeZfoRPsbCjih+xtMTOMGl4tzgLeG7CLgj8aHpc44iME/LWpEthEIQrAjNHY4qDKt/ET1Ehq4mhUZ9DWVx2az0M" } }, { "type": "step", "result": "=4\\sec^{4}\\left(θ\\right)-4\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vjDr4p6TvcgiSLV1uDo4U/jbMbEoNNbIR/7Q84OynCfTLx8mOdHYVzxX643JqKFIvTZkYD4gUeSoFrhGGhxq98kwYX4VRPb1geKkn+NU7mcE3SiHEhDlxIXJDVY+PPdvgQUxJPyUNnGfVirkcwpVO84fwxlvni/T8DIWnD0mw9U6FFV/0Hm6Y1hI7VaPzzKhkSoZP7CyArJlOo0FRny79Q==" } }, { "type": "step", "primary": "Add similar elements: $$2\\sec^{4}\\left(θ\\right)+4\\sec^{4}\\left(θ\\right)=6\\sec^{4}\\left(θ\\right)$$", "result": "=6\\sec^{4}\\left(θ\\right)-4\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7f48emNsIl4MI4hQZuZje4penFTHzh6X+DX3uvPdqYy7umM/GfBKdAtyuCOAeIKZmAJYpRu9XpYrd8NSAW2DdD0c/CXknyRKFbCevkP3vkuVyZXAw9UlCE0mEZtdsFPrV7q9+yf+iA4b58cx81CH0l/C30sSftAIFS6Qkpy19Ikp0x8irWzKvvppKMUvqTrnrUObZozYPTEmgGDR5V3grHDoUVX/QebpjWEjtVo/PMqGRKhk/sLICsmU6jQVGfLv1" } } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7xqDNX42OyJjPevbF+7IBng7708BkbVE1gb8/0kMEgzvXUIQv0aG2ncbDgSWX/DWycNZx/2EDub2dSsLw3NGg9xLC2T8NFZvTA+pff5AdXPdX9lRN4mlYq/zI0dgxEXGdPvdnhi0s3lZ+DJ7rEiTeqenxKTmAB1ahPFE542Dd/TeutOLi0kB5xHysaBsHIgntIaTy8gmKy5PY3el+RbUmlkmFP65WPUHCIGTpzT/EAccetg60+ZUlqP5jB+ReldUA5AMRxXWt+t9va0qZyagxyQ==" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7xI7dknI212svHgZHHIcoRvkdSragnZ5K8+WMRM5qjJ6ETkUamdUz59t0aHpV3uN0rE0Ib0rKQ2oM97vl/gDbowCWKUbvV6WK3fDUgFtg3Q+MfZ1BFRhhGnQf9q28g9nK/SzNF20Xrftd883pI0RALQ5+VkRmueXOP55Piqx5MyWBBTEk/JQ2cZ9WKuRzClU7m4RppR4RRNJR6+bqv/lHExUnakC9Y1jhr4uI97E9yvalDk6K3L00/zgNSSliALTtnpy3tAot7+5PIP9fBKZw2Llxun299aFzbjCyRFblUKE=" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{d^{6}}{dθ^{6}}\\left(-4\\sec^{2}\\left(θ\\right)+6\\sec^{4}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(-4\\sec^{2}\\left(θ\\right)+6\\sec^{4}\\left(θ\\right)\\right)=-8\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+24\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(-4\\sec^{2}\\left(θ\\right)+6\\sec^{4}\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=-\\frac{d}{dθ}\\left(4\\sec^{2}\\left(θ\\right)\\right)+\\frac{d}{dθ}\\left(6\\sec^{4}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(4\\sec^{2}\\left(θ\\right)\\right)=8\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(4\\sec^{2}\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=4\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Apply the chain rule:$${\\quad}2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{2},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{2}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{2}\\right)=2u$$", "input": "\\frac{d}{du}\\left(u^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2u^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2u", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYr+VZUwBnLdzbS6DZQ+f4s+k3hxk9aCfAWodBRxXgUexMchyqTAJWrzJaDbnNcFsJUeCBKuYKgaNJ253gLI69U79qbCA2QqVmvm3jGRXZ2ppvbGT4j1utMEkCDH25m/vlQ==" } }, { "type": "step", "result": "=2u\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgCSXIThcWH58oYoMPt18lzFqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidZClirlWNFPnHR1rvdyoFLI7IC+oqnwb1T6nPLpiw1eysw2NXPd8jZnr3fcPpqc7ejeh7+jKEzLb7VNCEMF3Z/Maswa5oIagKQFOfrnKm5+Gbb/PxX3awut5KxRxyZf8A=" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=4\\cdot\\:2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$4\\cdot\\:2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}8\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "4\\cdot\\:2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=8\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:2=8$$", "result": "=8\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{1+1}\\left(θ\\right)$$" ], "result": "=8\\sec^{1+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=8\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7NqBYpvGKZC5lMRQos2qph5K/zREn/U8meKQ3TvRjTQz8cZMsH3IX6lxp+ANHGbuCq47vuWedXv2WUg94ER8IwTfgzf+5OuhkULh+nunjrqae0Rv083m3K3ie3DVArOu08LfSxJ+0AgVLpCSnLX0iSqXiJoCK22T5YMsRbcSz8DsyFZaieHAvmtaw5r6RKguDHxgfR9uWXPlGtc+ysPJIG+IASZeFjDtawNGt9P21GJY=" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(6\\sec^{4}\\left(θ\\right)\\right)=24\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(6\\sec^{4}\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=6\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Apply the chain rule:$${\\quad}4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{4},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{4}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{4}\\right)=4u^{3}$$", "input": "\\frac{d}{du}\\left(u^{4}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=4u^{4-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=4u^{3}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYl3oEe4EC6n4B8Xgz0cKnkqk3hxk9aCfAWodBRxXgUexuzo0Fl6VhneutEteidn+Kv8//6/nV5O4fb8Xgwi7mapNFxUvwBeni+JEIFAdbegzYEOcQgyW4n51cR7AeMAm7Q==" } }, { "type": "step", "result": "=4u^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgA2CO5cm49xNEqFV0DMaZJRqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidVduWEnoDYDg06IbaTsp9TFtqiuNhFTvmF8ZCDqWZNoubfnOah8gAvLMKOeNMctC5tVYZ/3r82LxPimKUFXjQFStvD8GKDNJJN8qiu0XRcVZAYAvbRKqkRVDaDBHkSBtDwYHkfmHHTze3JXk3LjiJp" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=6\\cdot\\:4\\left(\\sec\\left(θ\\right)\\right)^{3}\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$6\\cdot\\:4\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}24\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "6\\cdot\\:4\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=24\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$6\\cdot\\:4=24$$", "result": "=24\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{3+1}\\left(θ\\right)$$" ], "result": "=24\\sec^{3+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$3+1=4$$", "result": "=24\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l3FkS70/4BFsebxcaZbHnTNSoP5XQt+n/5y371e8QHJROlTYN926lBwmiJdD5HUSzRqDxPUzBN6vjj5oJL9kUIdu2MPseykkfgn/Trt10KGp9ej8WEAztP3UNEC9n7jjZEt3ZXAiqUE0HIXrrrezJHZKl2mV4AlztiApTAFigz07AW6Jz0sXkPNp6b6xpHMqM1Kg/ldC36f/nLfvV7xActytFlzCV8oq55bHLwXbb0c=" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=-8\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+24\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{d^{5}}{dθ^{5}}\\left(-8\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+24\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(-8\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+24\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)=-8\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)+24\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(-8\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+24\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=-\\frac{d}{dθ}\\left(8\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+\\frac{d}{dθ}\\left(24\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(8\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\right)=8\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(8\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=8\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the Product Rule: $$\\left(f{\\cdot}g\\right)'=f'{\\cdot}g+f{\\cdot}g'$$", "secondary": [ "$$f=\\sec^{2}\\left(θ\\right),\\:g=\\tan\\left(θ\\right)$$" ], "result": "=8\\left(\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)\\tan\\left(θ\\right)+\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)\\sec^{2}\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Product%20Rule", "practiceTopic": "Product Rule" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{2},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{2}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{2}\\right)=2u$$", "input": "\\frac{d}{du}\\left(u^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2u^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2u", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYr+VZUwBnLdzbS6DZQ+f4s+k3hxk9aCfAWodBRxXgUexMchyqTAJWrzJaDbnNcFsJUeCBKuYKgaNJ253gLI69U79qbCA2QqVmvm3jGRXZ2ppvbGT4j1utMEkCDH25m/vlQ==" } }, { "type": "step", "result": "=2u\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgCSXIThcWH58oYoMPt18lzFqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidZClirlWNFPnHR1rvdyoFLI7IC+oqnwb1T6nPLpiw1eysw2NXPd8jZnr3fcPpqc7ejeh7+jKEzLb7VNCEMF3Z/Maswa5oIagKQFOfrnKm5+Gbb/PxX3awut5KxRxyZf8A=" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{1+1}\\left(θ\\right)$$" ], "result": "=2\\sec^{1+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7aY4rYekMZhUfP6sZ6mfqd6Lsz0lt1XLEr88Kc/cjJo5V00rpv8+ZC6TM10tVCSHspqrCcTwb/I0f8OwMjuU6eBAUa+AHsHd5OGpMiuFITjOuqvW3hAHH635V/vEEYIatHimBRYRqHSWeJkuUPhfTC5LQ0B1JNgsb5SXzbS+SCnAfGB9H25Zc+Ua1z7Kw8kgb4gBJl4WMO1rA0a30/bUYlg==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "result": "=\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYs0Vsx3c0nt53Utu5DgHTe3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK00HV/bLwwglXz/2YNx2i4JHggSrmCoGjSdud4CyOvVOgdQN0k+N3NHNAkEIoLrhB631k80nn+ODwI4ID0kCF5+z7zRz81SqRD02HEUE4Wqh" } }, { "type": "step", "result": "=8\\left(2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Simplify $$8\\left(2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\right):{\\quad}8\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)$$", "input": "8\\left(2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\right)", "result": "=8\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "$$2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)=2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)$$", "input": "2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\tan\\left(θ\\right)\\tan\\left(θ\\right)=\\:\\tan^{1+1}\\left(θ\\right)$$" ], "result": "=2\\sec^{2}\\left(θ\\right)\\tan^{1+1}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Md1SJKHYLxhOYRRbsLTu+axUZgt6FTmFSHnRsotkqemLeJKmeakO1+Q3aSIldG41s7BwSVdKEYSqn+4fKAQoGk0xmFIl1KTO5aAkhaHvKzBALhShbMGZIZsZWTO721HsWI4dG63F/aSz/PFDvu5FnO3sngYNzcydIedGQjjtM8sBMUhkLMqDD6KHvgZAgKMdRQzIIgXxnWeDhDTk2Lfum8RPUSGriaFRn0NZXHZrPQw=" } }, { "type": "interim", "title": "$$\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)=\\sec^{4}\\left(θ\\right)$$", "input": "\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)=\\:\\sec^{2+2}\\left(θ\\right)$$" ], "result": "=\\sec^{2+2}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+2=4$$", "result": "=\\sec^{4}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7AtrsReczPdUqIdn5/Ev5mFtNfVJqszzpMwg6spHO8F8gJ/ZZA32ZInFBpDtxBfiKn3Jyj2VJgBojKQHjINpkkETYvnjqwhFu6aWmDOYpAVMO/U6qyRZonn1WPQ/21pz3cKXVLLQXMot2gtnZH7qY5zkbLioo7YVO0SRnDAO/7Uokt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "=8\\left(\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Simplify $$2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{4}\\left(θ\\right):{\\quad}3\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)$$", "input": "2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{4}\\left(θ\\right)", "result": "=8\\left(3\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Rewrite using trig identities", "input": "\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)", "result": "=-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Use the Pythagorean identity: $$\\tan^{2}\\left(x\\right)+1=\\sec^{2}\\left(x\\right)$$", "secondary": [ "$$\\tan^{2}\\left(x\\right)=\\sec^{2}\\left(x\\right)-1$$" ], "result": "=\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right)" }, { "type": "interim", "title": "Simplify $$\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right):{\\quad}3\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)$$", "input": "\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right)", "result": "=3\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "interim", "title": "Expand $$2\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right):{\\quad}2\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)$$", "input": "2\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right)", "result": "=\\sec^{4}\\left(θ\\right)+2\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b-c\\right)=ab-ac$$", "secondary": [ "$$a=2\\sec^{2}\\left(θ\\right),\\:b=\\sec^{2}\\left(θ\\right),\\:c=1$$" ], "result": "=2\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)\\cdot\\:1", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "result": "=2\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)-2\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$2\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)-2\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right):{\\quad}2\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)$$", "input": "2\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)-2\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right)", "result": "=2\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "interim", "title": "$$2\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)=2\\sec^{4}\\left(θ\\right)$$", "input": "2\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)=\\:\\sec^{2+2}\\left(θ\\right)$$" ], "result": "=2\\sec^{2+2}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+2=4$$", "result": "=2\\sec^{4}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Md1SJKHYLxhOYRRbsLTu+dJYPDPDesonRI520G8zYRPehkKrn0era9rz8TlL+x/vyvn14PieOyuPP98CkiPIaXRfFgXB+vlRb/AHCJklpF+qQC22PXYzQL8KzAhJCzszqX/cSuhK5Ty2xqd/IdNr67A2LXxYB8f3C3KpWaLgll1U2fg2Eo8khoQfnK/P3ock" } }, { "type": "interim", "title": "$$2\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right)=2\\sec^{2}\\left(θ\\right)$$", "input": "2\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=2\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ht1jgkupea8+jUgT+M/VFPgkIqVBCzsTrV5i/ECNCw8tOtZYwUjyXhDTsNnn6Elr8U8lPA4O6WXezAN1J2P7Ue6vfsn/ogOG+fHMfNQh9Je7opJQAzFYPpqn/2bLavXutzgLeG7CLgj8aHpc44iMEx91+u64Nzpp2/441mWGcQ/ET1Ehq4mhUZ9DWVx2az0M" } }, { "type": "step", "result": "=2\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Md1SJKHYLxhOYRRbsLTu+fjbMbEoNNbIR/7Q84OynCfTLx8mOdHYVzxX643JqKFI1Jetp9AvoQcY2QSDwFJhz1kUvGSawlk7umQ8kurx5doE3SiHEhDlxIXJDVY+PPdvgQUxJPyUNnGfVirkcwpVO6RMkE3rByM/Y1Dxxfs3xFw6FFV/0Hm6Y1hI7VaPzzKhkSoZP7CyArJlOo0FRny79Q==" } }, { "type": "step", "primary": "Add similar elements: $$\\sec^{4}\\left(θ\\right)+2\\sec^{4}\\left(θ\\right)=3\\sec^{4}\\left(θ\\right)$$", "result": "=3\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ARkhlGZdazrdxNLzhlqePtRiXXTzenxbt00ahL9sDS5L0MjjUe01mdeSwRMoYPSBLTrWWMFI8l4Q07DZ5+hJazaxvejhvTJvqoS105+T9+xc3qH/hVuCay9igw/1IwZBNySLUGdPDJeW3uY1XSs479bA+zX4bD3u3gx65o2NJhP8KqIsxz0lwScBMkZS31XhRCRwG0yFaMyhS3uC5/uUeMvqafR00cq/gMXQBnztqPn0yTzBsNnlyCSr/zrcxdQG" } } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7FkSOkw7Nc+ISjlBZ7XiESO/dTLaj0YrNSw6+/ogyGX8k+jBZ4XxujMMeS3Qvsi6m7aEe0hvT9VtxXei653t2zQi9Vnn4imguIoWn3N1eqjdGIkb3QW2w7y4N83RSktbc/kUupyp7xXxB8YmKEQ5HoPs4SAUa1zTfp2gkDWnxoYVtNtNgAtA4xHxju0w39zFkZEt3ZXAiqUE0HIXrrrezJJi0cdkgnw2HPjRVpcHMxZbIciYVzjaAolEaNkVONtBBRppTNgMFons9bMkTzIHooA==" } } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7JzuoJyZ0ket+0by442o6nfkdSragnZ5K8+WMRM5qjJ6ETkUamdUz59t0aHpV3uN0rE0Ib0rKQ2oM97vl/gDbowCWKUbvV6WK3fDUgFtg3Q+nr1JqgQ+t+tehVxpQusgo+5k7zGGz9wmA2lyQsVWy+j6F2h9EjPuS9/S594y1Nd1FKk3fejFkyiOiq9iG9IkAIuSmJUDtLxRPyJ57Nkq7qjUyCLMvQZ782OaKpAVh7FOsVGYLehU5hUh50bKLZKnpo1d4a+pBCGznOTLxP2sni3enTnyZLeuZs6pBwuQMplUkt3WiGR7ZaCaXvz77bMjS" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(24\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)=24\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(24\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=24\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the Product Rule: $$\\left(f{\\cdot}g\\right)'=f'{\\cdot}g+f{\\cdot}g'$$", "secondary": [ "$$f=\\sec^{4}\\left(θ\\right),\\:g=\\tan\\left(θ\\right)$$" ], "result": "=24\\left(\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)\\tan\\left(θ\\right)+\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)\\sec^{4}\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Product%20Rule", "practiceTopic": "Product Rule" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)=4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{4},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{4}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{4}\\right)=4u^{3}$$", "input": "\\frac{d}{du}\\left(u^{4}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=4u^{4-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=4u^{3}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYl3oEe4EC6n4B8Xgz0cKnkqk3hxk9aCfAWodBRxXgUexuzo0Fl6VhneutEteidn+Kv8//6/nV5O4fb8Xgwi7mapNFxUvwBeni+JEIFAdbegzYEOcQgyW4n51cR7AeMAm7Q==" } }, { "type": "step", "result": "=4u^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgA2CO5cm49xNEqFV0DMaZJRqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidVduWEnoDYDg06IbaTsp9TFtqiuNhFTvmF8ZCDqWZNoubfnOah8gAvLMKOeNMctC5tVYZ/3r82LxPimKUFXjQFStvD8GKDNJJN8qiu0XRcVZAYAvbRKqkRVDaDBHkSBtDwYHkfmHHTze3JXk3LjiJp" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$4\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "4\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{3+1}\\left(θ\\right)$$" ], "result": "=4\\sec^{3+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$3+1=4$$", "result": "=4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7p9w76+/nvHitDrMIupHrrrPmAPxuc/RFHWXkB8A+mnqLeJKmeakO1+Q3aSIldG41PCyA7ixjCdfDYe+yac7KOvOIrkYdKW3Q58EnXaMU+7oslHPSvPf5UfEs5T+IFZ4XHjb2+5NLFZrsH9fcPWg/TXL71sHPAsAQtYZvjpaGSRwboK/Igc1IRC3EGruCoRzV1Zrc/SYRIR8JuxYpD+PJ+Q==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "result": "=\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYs0Vsx3c0nt53Utu5DgHTe3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK00HV/bLwwglXz/2YNx2i4JHggSrmCoGjSdud4CyOvVOgdQN0k+N3NHNAkEIoLrhB631k80nn+ODwI4ID0kCF5+z7zRz81SqRD02HEUE4Wqh" } }, { "type": "step", "result": "=24\\left(4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Simplify $$24\\left(4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right):{\\quad}24\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)$$", "input": "24\\left(4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right)", "result": "=24\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "$$4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)=4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)$$", "input": "4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\tan\\left(θ\\right)\\tan\\left(θ\\right)=\\:\\tan^{1+1}\\left(θ\\right)$$" ], "result": "=4\\sec^{4}\\left(θ\\right)\\tan^{1+1}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7GHerUg4zB8o0EeBO6BKV16xUZgt6FTmFSHnRsotkqemLeJKmeakO1+Q3aSIldG41PCyA7ixjCdfDYe+yac7KOvOIrkYdKW3Q58EnXaMU+7pALhShbMGZIZsZWTO721HsXaBT4KNKuhZZSFsFWnKRStdNJOkRdCtrWB4iMQVmMwBoAWE3NiWlLPrDNnXLOZLK3tKI9zGyi4xCTaccmboyHcRPUSGriaFRn0NZXHZrPQw=" } }, { "type": "interim", "title": "$$\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)=\\sec^{6}\\left(θ\\right)$$", "input": "\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)=\\:\\sec^{2+4}\\left(θ\\right)$$" ], "result": "=\\sec^{2+4}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+4=6$$", "result": "=\\sec^{6}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7AtrsReczPdUqIdn5/Ev5mLBxohAhN47qWD2nlwOHqfkgJ/ZZA32ZInFBpDtxBfiKn3Jyj2VJgBojKQHjINpkkG6JHhxn2McSW7OxLgQBi64O/U6qyRZonn1WPQ/21pz3xMe3sR8ZMoA2Ynlp2W853t3OcQiSW6rZ4zI81UhADI4kt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "=24\\left(\\sec^{6}\\left(θ\\right)+4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7jhDHfABKmSqcJJ1fAFAsGBiMJmjxigW8LortZKvrqEbWPF6oWuXggupsFvXrooIqxMe3sR8ZMoA2Ynlp2W853tMvHyY50dhXPFfrjcmooUg3CXpyb+Iu3cMrCn545PU8uO1NenZJP60hEwBH05kmlvOIrkYdKW3Q58EnXaMU+7r4ZAjrJM4mwvgfYovsxoopgQUxJPyUNnGfVirkcwpVOxKnUVO0TgjLT50Mmk0DzLYh0PvFonSnuZi7Z6ItIhmYhYbzXwn7iok10o2w+NJLRiZIb9CFuBJdPyi+MBoLCMd/LnYygZMYhjzpUCXiHwCy" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=-8\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)+24\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{d^{4}}{dθ^{4}}\\left(-8\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)+24\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(-8\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)+24\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)\\right)=-8\\left(-4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+24\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(-8\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)+24\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=-\\frac{d}{dθ}\\left(8\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)\\right)+\\frac{d}{dθ}\\left(24\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(8\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)\\right)=8\\left(-4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(8\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=8\\frac{d}{dθ}\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=8\\left(-\\frac{d}{dθ}\\left(2\\sec^{2}\\left(θ\\right)\\right)+\\frac{d}{dθ}\\left(3\\sec^{4}\\left(θ\\right)\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(2\\sec^{2}\\left(θ\\right)\\right)=4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(2\\sec^{2}\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=2\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Apply the chain rule:$${\\quad}2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{2},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{2}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{2}\\right)=2u$$", "input": "\\frac{d}{du}\\left(u^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2u^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2u", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYr+VZUwBnLdzbS6DZQ+f4s+k3hxk9aCfAWodBRxXgUexMchyqTAJWrzJaDbnNcFsJUeCBKuYKgaNJ253gLI69U79qbCA2QqVmvm3jGRXZ2ppvbGT4j1utMEkCDH25m/vlQ==" } }, { "type": "step", "result": "=2u\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgCSXIThcWH58oYoMPt18lzFqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidZClirlWNFPnHR1rvdyoFLI7IC+oqnwb1T6nPLpiw1eysw2NXPd8jZnr3fcPpqc7ejeh7+jKEzLb7VNCEMF3Z/Maswa5oIagKQFOfrnKm5+Gbb/PxX3awut5KxRxyZf8A=" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=2\\cdot\\:2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$2\\cdot\\:2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "2\\cdot\\:2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=4\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{1+1}\\left(θ\\right)$$" ], "result": "=4\\sec^{1+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s71jEHNygbGF2qQyJrTxQFvpK/zREn/U8meKQ3TvRjTQz8cZMsH3IX6lxp+ANHGbuCq47vuWedXv2WUg94ER8IwfulLd9uW7xFGDkqRxB3vSSe0Rv083m3K3ie3DVArOu08LfSxJ+0AgVLpCSnLX0iSnTHyKtbMq++mkoxS+pOuesyFZaieHAvmtaw5r6RKguDHxgfR9uWXPlGtc+ysPJIG+IASZeFjDtawNGt9P21GJY=" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(3\\sec^{4}\\left(θ\\right)\\right)=12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(3\\sec^{4}\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=3\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Apply the chain rule:$${\\quad}4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{4},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{4}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{4}\\right)=4u^{3}$$", "input": "\\frac{d}{du}\\left(u^{4}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=4u^{4-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=4u^{3}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYl3oEe4EC6n4B8Xgz0cKnkqk3hxk9aCfAWodBRxXgUexuzo0Fl6VhneutEteidn+Kv8//6/nV5O4fb8Xgwi7mapNFxUvwBeni+JEIFAdbegzYEOcQgyW4n51cR7AeMAm7Q==" } }, { "type": "step", "result": "=4u^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgA2CO5cm49xNEqFV0DMaZJRqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidVduWEnoDYDg06IbaTsp9TFtqiuNhFTvmF8ZCDqWZNoubfnOah8gAvLMKOeNMctC5tVYZ/3r82LxPimKUFXjQFStvD8GKDNJJN8qiu0XRcVZAYAvbRKqkRVDaDBHkSBtDwYHkfmHHTze3JXk3LjiJp" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=3\\cdot\\:4\\left(\\sec\\left(θ\\right)\\right)^{3}\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$3\\cdot\\:4\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "3\\cdot\\:4\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$3\\cdot\\:4=12$$", "result": "=12\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{3+1}\\left(θ\\right)$$" ], "result": "=12\\sec^{3+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$3+1=4$$", "result": "=12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7jFGb0q9vd+kdWvVYeHJHbDNSoP5XQt+n/5y371e8QHJROlTYN926lBwmiJdD5HUSzRqDxPUzBN6vjj5oJL9kUFPzsE52Mw7P00gxEE61byKp9ej8WEAztP3UNEC9n7jjZEt3ZXAiqUE0HIXrrrezJHZKl2mV4AlztiApTAFigz2gNkWvAT/c3O3uDO/qxcKwM1Kg/ldC36f/nLfvV7xActytFlzCV8oq55bHLwXbb0c=" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=8\\left(-4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(24\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)\\right)=24\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(24\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=24\\frac{d}{dθ}\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=24\\left(\\frac{d}{dθ}\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)+\\frac{d}{dθ}\\left(\\sec^{6}\\left(θ\\right)\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)=4\\left(4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=4\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the Product Rule: $$\\left(f{\\cdot}g\\right)'=f'{\\cdot}g+f{\\cdot}g'$$", "secondary": [ "$$f=\\sec^{4}\\left(θ\\right),\\:g=\\tan^{2}\\left(θ\\right)$$" ], "result": "=4\\left(\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)\\tan^{2}\\left(θ\\right)+\\frac{d}{dθ}\\left(\\tan^{2}\\left(θ\\right)\\right)\\sec^{4}\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Product%20Rule", "practiceTopic": "Product Rule" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)=4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{4},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{4}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{4}\\right)=4u^{3}$$", "input": "\\frac{d}{du}\\left(u^{4}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=4u^{4-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=4u^{3}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYl3oEe4EC6n4B8Xgz0cKnkqk3hxk9aCfAWodBRxXgUexuzo0Fl6VhneutEteidn+Kv8//6/nV5O4fb8Xgwi7mapNFxUvwBeni+JEIFAdbegzYEOcQgyW4n51cR7AeMAm7Q==" } }, { "type": "step", "result": "=4u^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgA2CO5cm49xNEqFV0DMaZJRqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidVduWEnoDYDg06IbaTsp9TFtqiuNhFTvmF8ZCDqWZNoubfnOah8gAvLMKOeNMctC5tVYZ/3r82LxPimKUFXjQFStvD8GKDNJJN8qiu0XRcVZAYAvbRKqkRVDaDBHkSBtDwYHkfmHHTze3JXk3LjiJp" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$4\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "4\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{3+1}\\left(θ\\right)$$" ], "result": "=4\\sec^{3+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$3+1=4$$", "result": "=4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7p9w76+/nvHitDrMIupHrrrPmAPxuc/RFHWXkB8A+mnqLeJKmeakO1+Q3aSIldG41PCyA7ixjCdfDYe+yac7KOvOIrkYdKW3Q58EnXaMU+7oslHPSvPf5UfEs5T+IFZ4XHjb2+5NLFZrsH9fcPWg/TXL71sHPAsAQtYZvjpaGSRwboK/Igc1IRC3EGruCoRzV1Zrc/SYRIR8JuxYpD+PJ+Q==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan^{2}\\left(θ\\right)\\right)=2\\tan\\left(θ\\right)\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan^{2}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}2\\tan\\left(θ\\right)\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan^{2}\\left(θ\\right)\\right)", "result": "=2\\tan\\left(θ\\right)\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{2},\\:\\:u=\\tan\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{2}\\right)\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{2}\\right)=2u$$", "input": "\\frac{d}{du}\\left(u^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2u^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2u", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYr+VZUwBnLdzbS6DZQ+f4s+k3hxk9aCfAWodBRxXgUexMchyqTAJWrzJaDbnNcFsJUeCBKuYKgaNJ253gLI69U79qbCA2QqVmvm3jGRXZ2ppvbGT4j1utMEkCDH25m/vlQ==" } }, { "type": "step", "result": "=2u\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\tan\\left(θ\\right)$$", "result": "=2\\tan\\left(θ\\right)\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYux7TmDJ3fopnHGBxoqO+paSXIThcWH58oYoMPt18lzFqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidIn+HfeW3QQwAkRa15J0NSAlj501WLIv46IdbBaParYow2V0WdtbiweShXv1KOwqCjeh7+jKEzLb7VNCEMF3Z/Maswa5oIagKQFOfrnKm5+Gbb/PxX3awut5KxRxyZf8A=" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "result": "=\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYs0Vsx3c0nt53Utu5DgHTe3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK00HV/bLwwglXz/2YNx2i4JHggSrmCoGjSdud4CyOvVOgdQN0k+N3NHNAkEIoLrhB631k80nn+ODwI4ID0kCF5+z7zRz81SqRD02HEUE4Wqh" } }, { "type": "step", "result": "=2\\tan\\left(θ\\right)\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=4\\left(4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{2}\\left(θ\\right)+2\\tan\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Simplify $$4\\left(4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{2}\\left(θ\\right)+2\\tan\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right):{\\quad}4\\left(4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)$$", "input": "4\\left(4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{2}\\left(θ\\right)+2\\tan\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right)", "result": "=4\\left(4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "$$4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{2}\\left(θ\\right)=4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)$$", "input": "4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\tan\\left(θ\\right)\\tan^{2}\\left(θ\\right)=\\:\\tan^{1+2}\\left(θ\\right)$$" ], "result": "=4\\sec^{4}\\left(θ\\right)\\tan^{1+2}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+2=3$$", "result": "=4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7GHerUg4zB8o0EeBO6BKV162cmBc6Js8szDUhC2Hxdqi5HNbiQL8+p2P+VJTsJbU7q47vuWedXv2WUg94ER8IwVADhNIEz39BpXewdhMCBb+bbDkw/o6UIkLjzolgTAsqkuHx4ulZqfxZHIsiQUpV1nyo9elyUFFpcPZW02cAt7EtbJdlgVP3iSyKFg6v1zm6cUwY1HWd9EJWOCb4O+kauMdM6yd/xAZrmYAkuRY8/rYkt3WiGR7ZaCaXvz77bMjS" } }, { "type": "interim", "title": "$$2\\tan\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)=2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "2\\tan\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)=\\:\\sec^{2+4}\\left(θ\\right)$$" ], "result": "=2\\tan\\left(θ\\right)\\sec^{2+4}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+4=6$$", "result": "=2\\tan\\left(θ\\right)\\sec^{6}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7OyotidIxGYYp2AND0rbhNaCGttAzkBWvgG18MKDLv3Rdbcltti6PzkTDSwkmBTY6q47vuWedXv2WUg94ER8IwSIYCdSNn6m2gNv8FkjgS1XBL9WNCz07kx2T4iGijNtQjT/+cTnIKvMqhzKaFBbxvzSei4QZxxrazp/cx7cYZvbVeYdTw2sAeZ9o0nz1uUihamteA8UyY/eLm9jagP57hkwLhsnE4D1C6IGVVz/pqlY=" } }, { "type": "step", "result": "=4\\left(2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)+4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7iza9NvP7nq2MIMeLHhROsvkdSragnZ5K8+WMRM5qjJ4/XhyePE1SthuGDJcaFMnncKXVLLQXMot2gtnZH7qY51d1u0XlM+NdfDNBhSESuvlV00rpv8+ZC6TM10tVCSHsi/RlR31DZv0142/jAcann2ASrhThz5UQhS79xvpIR8tIoCq5NivR6sCY2Sfa33kc8jzDW/n9HorFBpgomiqIe+5byrQDQVCXUD0vH/fvOdxyhd7tjiG+GxQNxDvGkZUlqqyy15Nuz77zqSegKeIT2iKI9Km/ZvzCnLF9eqmklKHqRJDJ/DeTzEayKJpmRlyWQi730E1b9JEo9956Qy3sPLLWww1IW5GUOlAGrcUL2OqwiNrEngO+NNvZ9sqNu+2V" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec^{6}\\left(θ\\right)\\right)=6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{6}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}6\\left(\\sec\\left(θ\\right)\\right)^{5}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{6}\\left(θ\\right)\\right)", "result": "=6\\left(\\sec\\left(θ\\right)\\right)^{5}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{6},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{6}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{6}\\right)=6u^{5}$$", "input": "\\frac{d}{du}\\left(u^{6}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=6u^{6-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=6u^{5}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYgOK76JOSNqGa1B1wU6ozWWk3hxk9aCfAWodBRxXgUexqSoDkuZ4Ab1St3bM8oqKcP8//6/nV5O4fb8Xgwi7mapNFxUvwBeni+JEIFAdbegzHfwjUTnSu4cWqw+8m79+NA==" } }, { "type": "step", "result": "=6u^{5}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=6\\left(\\sec\\left(θ\\right)\\right)^{5}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgBaJTGF5uPwSMzxoSyJNVZsqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKieMSe35zYqdvyntnEYxaQ88jJq7JZFtKgvxzzzuaqTGrebfnOah8gAvLMKOeNMctC5tVYZ/3r82LxPimKUFXjQFStvD8GKDNJJN8qiu0XRcVZAYAvbRKqkRVDaDBHkSBtDwYHkfmHHTze3JXk3LjiJp" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=6\\left(\\sec\\left(θ\\right)\\right)^{5}\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$6\\sec^{5}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "6\\sec^{5}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{5}\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{5+1}\\left(θ\\right)$$" ], "result": "=6\\sec^{5+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$5+1=6$$", "result": "=6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7QMJmgA2Sx2hYpGS5j/h1NLPmAPxuc/RFHWXkB8A+mnqLeJKmeakO1+Q3aSIldG417JiujyaxykLR7Tv0Nr+wOGprXgPFMmP3i5vY2oD+e4YslHPSvPf5UfEs5T+IFZ4XHjb2+5NLFZrsH9fcPWg/TRiIlnZT/wCWlhB0uSVNtvx/uR/EZqc6EbI3tnxSI5001Zrc/SYRIR8JuxYpD+PJ+Q==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=24\\left(4\\left(4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Expand $$4\\left(4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "4\\left(4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=24\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Expand $$4\\left(4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right):{\\quad}16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+8\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "4\\left(4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)", "result": "=16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+8\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)+6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b+c\\right)=ab+ac$$", "secondary": [ "$$a=4,\\:b=4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right),\\:c=2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)$$" ], "result": "=4\\cdot\\:4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+4\\cdot\\:2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "interim", "title": "Simplify $$4\\cdot\\:4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+4\\cdot\\:2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+8\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "4\\cdot\\:4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+4\\cdot\\:2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+8\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:4=16$$", "result": "=16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+4\\cdot\\:2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:2=8$$", "result": "=16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+8\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7iza9NvP7nq2MIMeLHhROsoL8UIMToMV6hDKJhY+Nk7toxrHhSJArgBpsnuAV9UCfBgOJ+lz5D5t+hsCNYCoDMHWD310L1+P2yDQQfMEhENFHx6CvpVa90MvMzklpzJWXZAzoThz1g+TITtf+WJ7KiyRRh7kCo0obc/CTaNyPGkYuuAKnMbBNnPJ2x/QM7ydV1sD7NfhsPe7eDHrmjY0mEyzoLVQPyp9R+5GC0prpwlf2K8DiYaJsjNsjOcdOcdQLW6BryIKCtddVLZsZyZ1dV9DmtpfPTl/o4ZJKfQIdsQqwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "primary": "Add similar elements: $$8\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)+6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)=14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7iza9NvP7nq2MIMeLHhROsoL8UIMToMV6hDKJhY+Nk7toxrHhSJArgBpsnuAV9UCf8SVGtBi1ovwgT5vDB2QfkF3B8WOxYrIgIdHvusvufLAAlilG71elit3w1IBbYN0PumuR+HimzGDTYbsfY1Q6JXyo9elyUFFpcPZW02cAt7EXYiLGBL7PG2xAqJd1w6BXKK1Q1tVX7SUdYWONKsZTX66q9beEAcfrflX+8QRghq2CcvL2jSzw1VlRcGX4ka6qwZ5KNwm43WeyBMBBKL30QvI8w1v5/R6KxQaYKJoqiHsLiDrnJdtlx1I2mhkkUTElN8b/k79QNO9t6sAoPBR3zJYojigZ8i1ymB3KwWBswnjiAEmXhYw7WsDRrfT9tRiW" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=-8\\left(-4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+24\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{d^{3}}{dθ^{3}}\\left(-8\\left(-4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+24\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(-8\\left(-4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+24\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)\\right)=-8\\left(-4\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)+12\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)\\right)+24\\left(64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+132\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+14\\sec^{8}\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(-8\\left(-4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+24\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=-\\frac{d}{dθ}\\left(8\\left(-4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)\\right)+\\frac{d}{dθ}\\left(24\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(8\\left(-4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)\\right)=8\\left(-4\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)+12\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(8\\left(-4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=8\\frac{d}{dθ}\\left(-4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=8\\left(-\\frac{d}{dθ}\\left(4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+\\frac{d}{dθ}\\left(12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\right)=4\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=4\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the Product Rule: $$\\left(f{\\cdot}g\\right)'=f'{\\cdot}g+f{\\cdot}g'$$", "secondary": [ "$$f=\\sec^{2}\\left(θ\\right),\\:g=\\tan\\left(θ\\right)$$" ], "result": "=4\\left(\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)\\tan\\left(θ\\right)+\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)\\sec^{2}\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Product%20Rule", "practiceTopic": "Product Rule" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{2},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{2}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{2}\\right)=2u$$", "input": "\\frac{d}{du}\\left(u^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2u^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2u", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYr+VZUwBnLdzbS6DZQ+f4s+k3hxk9aCfAWodBRxXgUexMchyqTAJWrzJaDbnNcFsJUeCBKuYKgaNJ253gLI69U79qbCA2QqVmvm3jGRXZ2ppvbGT4j1utMEkCDH25m/vlQ==" } }, { "type": "step", "result": "=2u\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgCSXIThcWH58oYoMPt18lzFqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidZClirlWNFPnHR1rvdyoFLI7IC+oqnwb1T6nPLpiw1eysw2NXPd8jZnr3fcPpqc7ejeh7+jKEzLb7VNCEMF3Z/Maswa5oIagKQFOfrnKm5+Gbb/PxX3awut5KxRxyZf8A=" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{1+1}\\left(θ\\right)$$" ], "result": "=2\\sec^{1+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7aY4rYekMZhUfP6sZ6mfqd6Lsz0lt1XLEr88Kc/cjJo5V00rpv8+ZC6TM10tVCSHspqrCcTwb/I0f8OwMjuU6eBAUa+AHsHd5OGpMiuFITjOuqvW3hAHH635V/vEEYIatHimBRYRqHSWeJkuUPhfTC5LQ0B1JNgsb5SXzbS+SCnAfGB9H25Zc+Ua1z7Kw8kgb4gBJl4WMO1rA0a30/bUYlg==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "result": "=\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYs0Vsx3c0nt53Utu5DgHTe3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK00HV/bLwwglXz/2YNx2i4JHggSrmCoGjSdud4CyOvVOgdQN0k+N3NHNAkEIoLrhB631k80nn+ODwI4ID0kCF5+z7zRz81SqRD02HEUE4Wqh" } }, { "type": "step", "result": "=4\\left(2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Simplify $$4\\left(2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\right):{\\quad}4\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)$$", "input": "4\\left(2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\right)", "result": "=4\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "$$2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)=2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)$$", "input": "2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\tan\\left(θ\\right)\\tan\\left(θ\\right)=\\:\\tan^{1+1}\\left(θ\\right)$$" ], "result": "=2\\sec^{2}\\left(θ\\right)\\tan^{1+1}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Md1SJKHYLxhOYRRbsLTu+axUZgt6FTmFSHnRsotkqemLeJKmeakO1+Q3aSIldG41s7BwSVdKEYSqn+4fKAQoGk0xmFIl1KTO5aAkhaHvKzBALhShbMGZIZsZWTO721HsWI4dG63F/aSz/PFDvu5FnO3sngYNzcydIedGQjjtM8sBMUhkLMqDD6KHvgZAgKMdRQzIIgXxnWeDhDTk2Lfum8RPUSGriaFRn0NZXHZrPQw=" } }, { "type": "interim", "title": "$$\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)=\\sec^{4}\\left(θ\\right)$$", "input": "\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)=\\:\\sec^{2+2}\\left(θ\\right)$$" ], "result": "=\\sec^{2+2}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+2=4$$", "result": "=\\sec^{4}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7AtrsReczPdUqIdn5/Ev5mFtNfVJqszzpMwg6spHO8F8gJ/ZZA32ZInFBpDtxBfiKn3Jyj2VJgBojKQHjINpkkETYvnjqwhFu6aWmDOYpAVMO/U6qyRZonn1WPQ/21pz3cKXVLLQXMot2gtnZH7qY5zkbLioo7YVO0SRnDAO/7Uokt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "=4\\left(\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Simplify $$2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{4}\\left(θ\\right):{\\quad}3\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)$$", "input": "2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{4}\\left(θ\\right)", "result": "=4\\left(3\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Rewrite using trig identities", "input": "\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)", "result": "=-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Use the Pythagorean identity: $$\\tan^{2}\\left(x\\right)+1=\\sec^{2}\\left(x\\right)$$", "secondary": [ "$$\\tan^{2}\\left(x\\right)=\\sec^{2}\\left(x\\right)-1$$" ], "result": "=\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right)" }, { "type": "interim", "title": "Simplify $$\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right):{\\quad}3\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)$$", "input": "\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right)", "result": "=3\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "interim", "title": "Expand $$2\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right):{\\quad}2\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)$$", "input": "2\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right)", "result": "=\\sec^{4}\\left(θ\\right)+2\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b-c\\right)=ab-ac$$", "secondary": [ "$$a=2\\sec^{2}\\left(θ\\right),\\:b=\\sec^{2}\\left(θ\\right),\\:c=1$$" ], "result": "=2\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)\\cdot\\:1", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "result": "=2\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)-2\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$2\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)-2\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right):{\\quad}2\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)$$", "input": "2\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)-2\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right)", "result": "=2\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "interim", "title": "$$2\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)=2\\sec^{4}\\left(θ\\right)$$", "input": "2\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)=\\:\\sec^{2+2}\\left(θ\\right)$$" ], "result": "=2\\sec^{2+2}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+2=4$$", "result": "=2\\sec^{4}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Md1SJKHYLxhOYRRbsLTu+dJYPDPDesonRI520G8zYRPehkKrn0era9rz8TlL+x/vyvn14PieOyuPP98CkiPIaXRfFgXB+vlRb/AHCJklpF+qQC22PXYzQL8KzAhJCzszqX/cSuhK5Ty2xqd/IdNr67A2LXxYB8f3C3KpWaLgll1U2fg2Eo8khoQfnK/P3ock" } }, { "type": "interim", "title": "$$2\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right)=2\\sec^{2}\\left(θ\\right)$$", "input": "2\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=2\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ht1jgkupea8+jUgT+M/VFPgkIqVBCzsTrV5i/ECNCw8tOtZYwUjyXhDTsNnn6Elr8U8lPA4O6WXezAN1J2P7Ue6vfsn/ogOG+fHMfNQh9Je7opJQAzFYPpqn/2bLavXutzgLeG7CLgj8aHpc44iMEx91+u64Nzpp2/441mWGcQ/ET1Ehq4mhUZ9DWVx2az0M" } }, { "type": "step", "result": "=2\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Md1SJKHYLxhOYRRbsLTu+fjbMbEoNNbIR/7Q84OynCfTLx8mOdHYVzxX643JqKFI1Jetp9AvoQcY2QSDwFJhz1kUvGSawlk7umQ8kurx5doE3SiHEhDlxIXJDVY+PPdvgQUxJPyUNnGfVirkcwpVO6RMkE3rByM/Y1Dxxfs3xFw6FFV/0Hm6Y1hI7VaPzzKhkSoZP7CyArJlOo0FRny79Q==" } }, { "type": "step", "primary": "Add similar elements: $$\\sec^{4}\\left(θ\\right)+2\\sec^{4}\\left(θ\\right)=3\\sec^{4}\\left(θ\\right)$$", "result": "=3\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ARkhlGZdazrdxNLzhlqePtRiXXTzenxbt00ahL9sDS5L0MjjUe01mdeSwRMoYPSBLTrWWMFI8l4Q07DZ5+hJazaxvejhvTJvqoS105+T9+xc3qH/hVuCay9igw/1IwZBNySLUGdPDJeW3uY1XSs479bA+zX4bD3u3gx65o2NJhP8KqIsxz0lwScBMkZS31XhRCRwG0yFaMyhS3uC5/uUeMvqafR00cq/gMXQBnztqPn0yTzBsNnlyCSr/zrcxdQG" } } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7FkSOkw7Nc+ISjlBZ7XiESO/dTLaj0YrNSw6+/ogyGX8k+jBZ4XxujMMeS3Qvsi6m7aEe0hvT9VtxXei653t2zQi9Vnn4imguIoWn3N1eqjdGIkb3QW2w7y4N83RSktbc/kUupyp7xXxB8YmKEQ5HoPs4SAUa1zTfp2gkDWnxoYVtNtNgAtA4xHxju0w39zFkZEt3ZXAiqUE0HIXrrrezJJi0cdkgnw2HPjRVpcHMxZbIciYVzjaAolEaNkVONtBBRppTNgMFons9bMkTzIHooA==" } } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7aTuTexD/CTl68iqGJang3fkdSragnZ5K8+WMRM5qjJ6ETkUamdUz59t0aHpV3uN0rE0Ib0rKQ2oM97vl/gDbowCWKUbvV6WK3fDUgFtg3Q8XyqIsLZjT7Df8hUObtCZb+5k7zGGz9wmA2lyQsVWy+j6F2h9EjPuS9/S594y1Nd1FKk3fejFkyiOiq9iG9IkAIuSmJUDtLxRPyJ57Nkq7qlZOK6n0rXJ0Nc64aGIzHiSsVGYLehU5hUh50bKLZKnpo1d4a+pBCGznOTLxP2sni3enTnyZLeuZs6pBwuQMplUkt3WiGR7ZaCaXvz77bMjS" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)=12\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=12\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the Product Rule: $$\\left(f{\\cdot}g\\right)'=f'{\\cdot}g+f{\\cdot}g'$$", "secondary": [ "$$f=\\sec^{4}\\left(θ\\right),\\:g=\\tan\\left(θ\\right)$$" ], "result": "=12\\left(\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)\\tan\\left(θ\\right)+\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)\\sec^{4}\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Product%20Rule", "practiceTopic": "Product Rule" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)=4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{4},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{4}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{4}\\right)=4u^{3}$$", "input": "\\frac{d}{du}\\left(u^{4}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=4u^{4-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=4u^{3}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYl3oEe4EC6n4B8Xgz0cKnkqk3hxk9aCfAWodBRxXgUexuzo0Fl6VhneutEteidn+Kv8//6/nV5O4fb8Xgwi7mapNFxUvwBeni+JEIFAdbegzYEOcQgyW4n51cR7AeMAm7Q==" } }, { "type": "step", "result": "=4u^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgA2CO5cm49xNEqFV0DMaZJRqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidVduWEnoDYDg06IbaTsp9TFtqiuNhFTvmF8ZCDqWZNoubfnOah8gAvLMKOeNMctC5tVYZ/3r82LxPimKUFXjQFStvD8GKDNJJN8qiu0XRcVZAYAvbRKqkRVDaDBHkSBtDwYHkfmHHTze3JXk3LjiJp" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$4\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "4\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{3+1}\\left(θ\\right)$$" ], "result": "=4\\sec^{3+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$3+1=4$$", "result": "=4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7p9w76+/nvHitDrMIupHrrrPmAPxuc/RFHWXkB8A+mnqLeJKmeakO1+Q3aSIldG41PCyA7ixjCdfDYe+yac7KOvOIrkYdKW3Q58EnXaMU+7oslHPSvPf5UfEs5T+IFZ4XHjb2+5NLFZrsH9fcPWg/TXL71sHPAsAQtYZvjpaGSRwboK/Igc1IRC3EGruCoRzV1Zrc/SYRIR8JuxYpD+PJ+Q==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "result": "=\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYs0Vsx3c0nt53Utu5DgHTe3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK00HV/bLwwglXz/2YNx2i4JHggSrmCoGjSdud4CyOvVOgdQN0k+N3NHNAkEIoLrhB631k80nn+ODwI4ID0kCF5+z7zRz81SqRD02HEUE4Wqh" } }, { "type": "step", "result": "=12\\left(4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Simplify $$12\\left(4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right):{\\quad}12\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)$$", "input": "12\\left(4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right)", "result": "=12\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "$$4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)=4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)$$", "input": "4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\tan\\left(θ\\right)\\tan\\left(θ\\right)=\\:\\tan^{1+1}\\left(θ\\right)$$" ], "result": "=4\\sec^{4}\\left(θ\\right)\\tan^{1+1}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7GHerUg4zB8o0EeBO6BKV16xUZgt6FTmFSHnRsotkqemLeJKmeakO1+Q3aSIldG41PCyA7ixjCdfDYe+yac7KOvOIrkYdKW3Q58EnXaMU+7pALhShbMGZIZsZWTO721HsXaBT4KNKuhZZSFsFWnKRStdNJOkRdCtrWB4iMQVmMwBoAWE3NiWlLPrDNnXLOZLK3tKI9zGyi4xCTaccmboyHcRPUSGriaFRn0NZXHZrPQw=" } }, { "type": "interim", "title": "$$\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)=\\sec^{6}\\left(θ\\right)$$", "input": "\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)=\\:\\sec^{2+4}\\left(θ\\right)$$" ], "result": "=\\sec^{2+4}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+4=6$$", "result": "=\\sec^{6}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7AtrsReczPdUqIdn5/Ev5mLBxohAhN47qWD2nlwOHqfkgJ/ZZA32ZInFBpDtxBfiKn3Jyj2VJgBojKQHjINpkkG6JHhxn2McSW7OxLgQBi64O/U6qyRZonn1WPQ/21pz3xMe3sR8ZMoA2Ynlp2W853t3OcQiSW6rZ4zI81UhADI4kt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "=12\\left(\\sec^{6}\\left(θ\\right)+4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7lv52tOJZUIL6vqIybdOpThiMJmjxigW8LortZKvrqEbWPF6oWuXggupsFvXrooIqxMe3sR8ZMoA2Ynlp2W853tMvHyY50dhXPFfrjcmooUinnaTA6rKFuYRlEeyjdJSquO1NenZJP60hEwBH05kmlvOIrkYdKW3Q58EnXaMU+7r4ZAjrJM4mwvgfYovsxoopgQUxJPyUNnGfVirkcwpVO48SyOdzKmRYNaCIczIhnoIh0PvFonSnuZi7Z6ItIhmYhYbzXwn7iok10o2w+NJLRiZIb9CFuBJdPyi+MBoLCMd/LnYygZMYhjzpUCXiHwCy" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=8\\left(-4\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)+12\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(24\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)\\right)=24\\left(64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+132\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+14\\sec^{8}\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(24\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=24\\frac{d}{dθ}\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=24\\left(\\frac{d}{dθ}\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)\\right)+\\frac{d}{dθ}\\left(14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)\\right)=16\\left(4\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+3\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=16\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the Product Rule: $$\\left(f{\\cdot}g\\right)'=f'{\\cdot}g+f{\\cdot}g'$$", "secondary": [ "$$f=\\sec^{4}\\left(θ\\right),\\:g=\\tan^{3}\\left(θ\\right)$$" ], "result": "=16\\left(\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)\\tan^{3}\\left(θ\\right)+\\frac{d}{dθ}\\left(\\tan^{3}\\left(θ\\right)\\right)\\sec^{4}\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Product%20Rule", "practiceTopic": "Product Rule" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)=4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{4},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{4}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{4}\\right)=4u^{3}$$", "input": "\\frac{d}{du}\\left(u^{4}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=4u^{4-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=4u^{3}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYl3oEe4EC6n4B8Xgz0cKnkqk3hxk9aCfAWodBRxXgUexuzo0Fl6VhneutEteidn+Kv8//6/nV5O4fb8Xgwi7mapNFxUvwBeni+JEIFAdbegzYEOcQgyW4n51cR7AeMAm7Q==" } }, { "type": "step", "result": "=4u^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgA2CO5cm49xNEqFV0DMaZJRqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidVduWEnoDYDg06IbaTsp9TFtqiuNhFTvmF8ZCDqWZNoubfnOah8gAvLMKOeNMctC5tVYZ/3r82LxPimKUFXjQFStvD8GKDNJJN8qiu0XRcVZAYAvbRKqkRVDaDBHkSBtDwYHkfmHHTze3JXk3LjiJp" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$4\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "4\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{3+1}\\left(θ\\right)$$" ], "result": "=4\\sec^{3+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$3+1=4$$", "result": "=4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7p9w76+/nvHitDrMIupHrrrPmAPxuc/RFHWXkB8A+mnqLeJKmeakO1+Q3aSIldG41PCyA7ixjCdfDYe+yac7KOvOIrkYdKW3Q58EnXaMU+7oslHPSvPf5UfEs5T+IFZ4XHjb2+5NLFZrsH9fcPWg/TXL71sHPAsAQtYZvjpaGSRwboK/Igc1IRC3EGruCoRzV1Zrc/SYRIR8JuxYpD+PJ+Q==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan^{3}\\left(θ\\right)\\right)=3\\tan^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan^{3}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}3\\left(\\tan\\left(θ\\right)\\right)^{2}\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan^{3}\\left(θ\\right)\\right)", "result": "=3\\left(\\tan\\left(θ\\right)\\right)^{2}\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{3},\\:\\:u=\\tan\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{3}\\right)\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{3}\\right)=3u^{2}$$", "input": "\\frac{d}{du}\\left(u^{3}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=3u^{3-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=3u^{2}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYjNrXZy15fg+DDm4IZ/kmJKk3hxk9aCfAWodBRxXgUexYrhQEWwJJMG//0JI2CMZRv8//6/nV5O4fb8Xgwi7mapNFxUvwBeni+JEIFAdbegz8nAGF3XMCO2i5cp0uYL6iA==" } }, { "type": "step", "result": "=3u^{2}\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\tan\\left(θ\\right)$$", "result": "=3\\left(\\tan\\left(θ\\right)\\right)^{2}\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYux7TmDJ3fopnHGBxoqO+pYseUrH1bTMvpsf9SfQ9YpVqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKifDI4IHJ+buH5M+0PHLeGB4Be4N1NohQPDRW/BmsNJaPsq1KfCFKwSwA0qmWovUVP9tVYZ/3r82LxPimKUFXjQFStvD8GKDNJJN8qiu0XRcVZAYAvbRKqkRVDaDBHkSBtDwYHkfmHHTze3JXk3LjiJp" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "result": "=\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYs0Vsx3c0nt53Utu5DgHTe3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK00HV/bLwwglXz/2YNx2i4JHggSrmCoGjSdud4CyOvVOgdQN0k+N3NHNAkEIoLrhB631k80nn+ODwI4ID0kCF5+z7zRz81SqRD02HEUE4Wqh" } }, { "type": "step", "result": "=3\\left(\\tan\\left(θ\\right)\\right)^{2}\\sec^{2}\\left(θ\\right)" }, { "type": "step", "primary": "Simplify", "result": "=3\\tan^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=16\\left(4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{3}\\left(θ\\right)+3\\tan^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Simplify $$16\\left(4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{3}\\left(θ\\right)+3\\tan^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right):{\\quad}16\\left(4\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+3\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)$$", "input": "16\\left(4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{3}\\left(θ\\right)+3\\tan^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right)", "result": "=16\\left(4\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+3\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "$$4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{3}\\left(θ\\right)=4\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)$$", "input": "4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{3}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\tan\\left(θ\\right)\\tan^{3}\\left(θ\\right)=\\:\\tan^{1+3}\\left(θ\\right)$$" ], "result": "=4\\sec^{4}\\left(θ\\right)\\tan^{1+3}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+3=4$$", "result": "=4\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7GHerUg4zB8o0EeBO6BKV162cmBc6Js8szDUhC2HxdqjUgkMhq/lXB9qI3pCS5SJMq47vuWedXv2WUg94ER8IwVADhNIEz39BpXewdhMCBb8aab3UY3H06dpMJimNIyD5kuHx4ulZqfxZHIsiQUpV1nyo9elyUFFpcPZW02cAt7EdWYTaqPpaDtAHPW/SgMNicUwY1HWd9EJWOCb4O+kauAND0fAz408eCOOnBAkZ69Akt3WiGR7ZaCaXvz77bMjS" } }, { "type": "interim", "title": "$$3\\tan^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)=3\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)$$", "input": "3\\tan^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)=\\:\\sec^{2+4}\\left(θ\\right)$$" ], "result": "=3\\tan^{2}\\left(θ\\right)\\sec^{2+4}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+4=6$$", "result": "=3\\tan^{2}\\left(θ\\right)\\sec^{6}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7xud7SC9MT1MQdbTGBqrMw+H0qCTik4j9uS7jTF7ng2/8DCHl2/1cUFH2kQZvGLuVzRqDxPUzBN6vjj5oJL9kUEngkySEzEl6diiSTBzPnLnWxcLMxZwB9cGBLGbwbyjTrqr1t4QBx+t+Vf7xBGCGrW9B4jZ4XSgntIkqjxqF2Bvh9Kgk4pOI/bku40xe54Nv6RICywOwLOWusKkGpuhUtYx6/teVLREoXphSPTLZ2MbET1Ehq4mhUZ9DWVx2az0M" } }, { "type": "step", "result": "=16\\left(3\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+4\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7C43rUQmr5KAYI3UGEng6yRiMJmjxigW8LortZKvrqEYERpCXXFEE4hLCZfTeroF6V7JvkhZ2EFhPsvL8iYbkgPpadcGORE5bHX37Io3viWolp/MB1OadflBmLQoq5Pmlo5FYteSPKwXny4uCMrdsKwS9KcmQjztCcdjD5gRjJOz1a03PQ0UrDYBem+/aln9X84iuRh0pbdDnwSddoxT7uh96fmgcT2t23FrnniYZ1CmBBTEk/JQ2cZ9WKuRzClU7UxxUEaCmP0MbiSArdWRj6iHQ+8WidKe5mLtnoi0iGZhpK7ceo/th4B1fP0TRnZr9A1c/UDX6QxYlZyg/KYfMkDSei4QZxxrazp/cx7cYZvYA0hWLB/nt1m4ggrh+mxlbvzIPeEtDfcHv/z8uls8Teg==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)=14\\left(6\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{8}\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=14\\frac{d}{dθ}\\left(\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the Product Rule: $$\\left(f{\\cdot}g\\right)'=f'{\\cdot}g+f{\\cdot}g'$$", "secondary": [ "$$f=\\sec^{6}\\left(θ\\right),\\:g=\\tan\\left(θ\\right)$$" ], "result": "=14\\left(\\frac{d}{dθ}\\left(\\sec^{6}\\left(θ\\right)\\right)\\tan\\left(θ\\right)+\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)\\sec^{6}\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Product%20Rule", "practiceTopic": "Product Rule" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec^{6}\\left(θ\\right)\\right)=6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{6}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}6\\left(\\sec\\left(θ\\right)\\right)^{5}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{6}\\left(θ\\right)\\right)", "result": "=6\\left(\\sec\\left(θ\\right)\\right)^{5}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{6},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{6}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{6}\\right)=6u^{5}$$", "input": "\\frac{d}{du}\\left(u^{6}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=6u^{6-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=6u^{5}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYgOK76JOSNqGa1B1wU6ozWWk3hxk9aCfAWodBRxXgUexqSoDkuZ4Ab1St3bM8oqKcP8//6/nV5O4fb8Xgwi7mapNFxUvwBeni+JEIFAdbegzHfwjUTnSu4cWqw+8m79+NA==" } }, { "type": "step", "result": "=6u^{5}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=6\\left(\\sec\\left(θ\\right)\\right)^{5}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgBaJTGF5uPwSMzxoSyJNVZsqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKieMSe35zYqdvyntnEYxaQ88jJq7JZFtKgvxzzzuaqTGrebfnOah8gAvLMKOeNMctC5tVYZ/3r82LxPimKUFXjQFStvD8GKDNJJN8qiu0XRcVZAYAvbRKqkRVDaDBHkSBtDwYHkfmHHTze3JXk3LjiJp" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=6\\left(\\sec\\left(θ\\right)\\right)^{5}\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$6\\sec^{5}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "6\\sec^{5}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{5}\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{5+1}\\left(θ\\right)$$" ], "result": "=6\\sec^{5+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$5+1=6$$", "result": "=6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7QMJmgA2Sx2hYpGS5j/h1NLPmAPxuc/RFHWXkB8A+mnqLeJKmeakO1+Q3aSIldG417JiujyaxykLR7Tv0Nr+wOGprXgPFMmP3i5vY2oD+e4YslHPSvPf5UfEs5T+IFZ4XHjb2+5NLFZrsH9fcPWg/TRiIlnZT/wCWlhB0uSVNtvx/uR/EZqc6EbI3tnxSI5001Zrc/SYRIR8JuxYpD+PJ+Q==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "result": "=\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYs0Vsx3c0nt53Utu5DgHTe3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK00HV/bLwwglXz/2YNx2i4JHggSrmCoGjSdud4CyOvVOgdQN0k+N3NHNAkEIoLrhB631k80nn+ODwI4ID0kCF5+z7zRz81SqRD02HEUE4Wqh" } }, { "type": "step", "result": "=14\\left(6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{6}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Simplify $$14\\left(6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{6}\\left(θ\\right)\\right):{\\quad}14\\left(6\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{8}\\left(θ\\right)\\right)$$", "input": "14\\left(6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{6}\\left(θ\\right)\\right)", "result": "=14\\left(6\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{8}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "$$6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)=6\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)$$", "input": "6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\tan\\left(θ\\right)\\tan\\left(θ\\right)=\\:\\tan^{1+1}\\left(θ\\right)$$" ], "result": "=6\\sec^{6}\\left(θ\\right)\\tan^{1+1}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=6\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7H4jwT8QhunkiDAGNb+RcvaxUZgt6FTmFSHnRsotkqemLeJKmeakO1+Q3aSIldG417JiujyaxykLR7Tv0Nr+wOGprXgPFMmP3i5vY2oD+e4ZALhShbMGZIZsZWTO721Hsn1G20l7DmbyzInDIeLRU/ZYojigZ8i1ymB3KwWBswnjgWFmp7EeEDyNH2q8Tm1k1jHr+15UtEShemFI9MtnYxsRPUSGriaFRn0NZXHZrPQw=" } }, { "type": "interim", "title": "$$\\sec^{2}\\left(θ\\right)\\sec^{6}\\left(θ\\right)=\\sec^{8}\\left(θ\\right)$$", "input": "\\sec^{2}\\left(θ\\right)\\sec^{6}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{2}\\left(θ\\right)\\sec^{6}\\left(θ\\right)=\\:\\sec^{2+6}\\left(θ\\right)$$" ], "result": "=\\sec^{2+6}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+6=8$$", "result": "=\\sec^{8}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7AtrsReczPdUqIdn5/Ev5mH+O3FtTTH45VMxFH8DJ4BEgJ/ZZA32ZInFBpDtxBfiKn3Jyj2VJgBojKQHjINpkkI6d/oVrekw7T/Cky7x1KxcO/U6qyRZonn1WPQ/21pz3HK+miBBOGkRkZzIN3nn5xFhBmsSvLw1yh3wbSiSoUDgkt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "=14\\left(\\sec^{8}\\left(θ\\right)+6\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7nO0mFYRqkI6v7Sy9VTWwjRiMJmjxigW8LortZKvrqEbWPF6oWuXggupsFvXrooIqHK+miBBOGkRkZzIN3nn5xNMvHyY50dhXPFfrjcmooUiPqGSxzi9gRanHsLmpQgKwsfoApmSO56Ak/1ybzTsFLmprXgPFMmP3i5vY2oD+e4b4ZAjrJM4mwvgfYovsxoopgQUxJPyUNnGfVirkcwpVO2TUlQSCHNJw/4wJ8m9ormYqoLNuWbSjE1JT+hrPzX/RhYbzXwn7iok10o2w+NJLRiZIb9CFuBJdPyi+MBoLCMeBkcS/lgaNKtB6zcW1E3BB" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=24\\left(16\\left(4\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+3\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)+14\\left(6\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{8}\\left(θ\\right)\\right)\\right)" }, { "type": "interim", "title": "Expand $$16\\left(4\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+3\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)+14\\left(6\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{8}\\left(θ\\right)\\right):{\\quad}64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+132\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+14\\sec^{8}\\left(θ\\right)$$", "input": "16\\left(4\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+3\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)+14\\left(6\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{8}\\left(θ\\right)\\right)", "result": "=24\\left(64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+132\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+14\\sec^{8}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Expand $$16\\left(4\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+3\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right):{\\quad}64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+48\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)$$", "input": "16\\left(4\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+3\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)", "result": "=64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+48\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+14\\left(6\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{8}\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b+c\\right)=ab+ac$$", "secondary": [ "$$a=16,\\:b=4\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right),\\:c=3\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)$$" ], "result": "=16\\cdot\\:4\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+16\\cdot\\:3\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "interim", "title": "Simplify $$16\\cdot\\:4\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+16\\cdot\\:3\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right):{\\quad}64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+48\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)$$", "input": "16\\cdot\\:4\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+16\\cdot\\:3\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)", "result": "=64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+48\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$16\\cdot\\:4=64$$", "result": "=64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+16\\cdot\\:3\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)" }, { "type": "step", "primary": "Multiply the numbers: $$16\\cdot\\:3=48$$", "result": "=64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+48\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7C43rUQmr5KAYI3UGEng6yV6kSs12eASfWmxynRkYZ3imWJLfEdyBmZh9L6HIlMnFSPKgibEOx6xAH14rsVQdDd6GQqufR6tr2vPxOUv7H+/wV+6wolBQOY9LPugwYJ6C3tKI9zGyi4xCTaccmboyHVWGy3JBiLa3A/i0ciCokDrRsdR53zGMNMouUmBvz53xupxJf7boNv1S9Hc0DXJXvt7UD16RqFQha281as3i/3gIhVb1SjznyQ5krYRiFqNmoU0M/hfYOSS/FiVCoXaNrXxGVgveSDjBoBB0CLM8z96ArlFiGY8qAdw44hDradHvvzIPeEtDfcHv/z8uls8Teg==" } }, { "type": "interim", "title": "Expand $$14\\left(6\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{8}\\left(θ\\right)\\right):{\\quad}84\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+14\\sec^{8}\\left(θ\\right)$$", "input": "14\\left(6\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{8}\\left(θ\\right)\\right)", "result": "=64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+48\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+84\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+14\\sec^{8}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b+c\\right)=ab+ac$$", "secondary": [ "$$a=14,\\:b=6\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right),\\:c=\\sec^{8}\\left(θ\\right)$$" ], "result": "=14\\cdot\\:6\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+14\\sec^{8}\\left(θ\\right)", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "primary": "Multiply the numbers: $$14\\cdot\\:6=84$$", "result": "=84\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+14\\sec^{8}\\left(θ\\right)" } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7nO0mFYRqkI6v7Sy9VTWwjSvxearhctbAPxX3QGBh5kkx0tWj1vW7tIvP14Mwoy6mLTrWWMFI8l4Q07DZ5+hJazPkBMIuMrWSeavgWBoF37pgEq4U4c+VEIUu/cb6SEfLgBHHd6NlsvdNQCpoDejgKWxRFK5IwscCsNiX9YX0G6d6pfF1z6umzUJTJvt+ojYZ00PxkCV/xFqV0mmm+Nwrth1Y04xtNK2BCM2tE3PxmDme5tn70JIT2jKyMskERPvitGNYJGuT8kFILfu1szjq+A==" } }, { "type": "step", "primary": "Add similar elements: $$48\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+84\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)=132\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)$$", "result": "=64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+132\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+14\\sec^{8}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=-8\\left(-4\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)+12\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)\\right)+24\\left(64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+132\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+14\\sec^{8}\\left(θ\\right)\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{d^{2}}{dθ^{2}}\\left(-8\\left(-4\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)+12\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)\\right)+24\\left(64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+132\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+14\\sec^{8}\\left(θ\\right)\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(-8\\left(-4\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)+12\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)\\right)+24\\left(64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+132\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+14\\sec^{8}\\left(θ\\right)\\right)\\right)=-8\\left(-4\\left(-4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+12\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)\\right)+24\\left(256\\tan^{5}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+1048\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+376\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(-8\\left(-4\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)+12\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)\\right)+24\\left(64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+132\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+14\\sec^{8}\\left(θ\\right)\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=-\\frac{d}{dθ}\\left(8\\left(-4\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)+12\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)\\right)\\right)+\\frac{d}{dθ}\\left(24\\left(64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+132\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+14\\sec^{8}\\left(θ\\right)\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(8\\left(-4\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)+12\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)\\right)\\right)=8\\left(-4\\left(-4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+12\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(8\\left(-4\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)+12\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=8\\frac{d}{dθ}\\left(-4\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)+12\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)\\right)" }, { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=8\\left(-\\frac{d}{dθ}\\left(4\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)\\right)+\\frac{d}{dθ}\\left(12\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(4\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)\\right)=4\\left(-4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(4\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=4\\frac{d}{dθ}\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=4\\left(-\\frac{d}{dθ}\\left(2\\sec^{2}\\left(θ\\right)\\right)+\\frac{d}{dθ}\\left(3\\sec^{4}\\left(θ\\right)\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(2\\sec^{2}\\left(θ\\right)\\right)=4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(2\\sec^{2}\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=2\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Apply the chain rule:$${\\quad}2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{2},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{2}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{2}\\right)=2u$$", "input": "\\frac{d}{du}\\left(u^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2u^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2u", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYr+VZUwBnLdzbS6DZQ+f4s+k3hxk9aCfAWodBRxXgUexMchyqTAJWrzJaDbnNcFsJUeCBKuYKgaNJ253gLI69U79qbCA2QqVmvm3jGRXZ2ppvbGT4j1utMEkCDH25m/vlQ==" } }, { "type": "step", "result": "=2u\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgCSXIThcWH58oYoMPt18lzFqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidZClirlWNFPnHR1rvdyoFLI7IC+oqnwb1T6nPLpiw1eysw2NXPd8jZnr3fcPpqc7ejeh7+jKEzLb7VNCEMF3Z/Maswa5oIagKQFOfrnKm5+Gbb/PxX3awut5KxRxyZf8A=" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=2\\cdot\\:2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$2\\cdot\\:2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "2\\cdot\\:2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=4\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{1+1}\\left(θ\\right)$$" ], "result": "=4\\sec^{1+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s71jEHNygbGF2qQyJrTxQFvpK/zREn/U8meKQ3TvRjTQz8cZMsH3IX6lxp+ANHGbuCq47vuWedXv2WUg94ER8IwfulLd9uW7xFGDkqRxB3vSSe0Rv083m3K3ie3DVArOu08LfSxJ+0AgVLpCSnLX0iSnTHyKtbMq++mkoxS+pOuesyFZaieHAvmtaw5r6RKguDHxgfR9uWXPlGtc+ysPJIG+IASZeFjDtawNGt9P21GJY=" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(3\\sec^{4}\\left(θ\\right)\\right)=12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(3\\sec^{4}\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=3\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Apply the chain rule:$${\\quad}4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{4},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{4}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{4}\\right)=4u^{3}$$", "input": "\\frac{d}{du}\\left(u^{4}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=4u^{4-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=4u^{3}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYl3oEe4EC6n4B8Xgz0cKnkqk3hxk9aCfAWodBRxXgUexuzo0Fl6VhneutEteidn+Kv8//6/nV5O4fb8Xgwi7mapNFxUvwBeni+JEIFAdbegzYEOcQgyW4n51cR7AeMAm7Q==" } }, { "type": "step", "result": "=4u^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgA2CO5cm49xNEqFV0DMaZJRqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidVduWEnoDYDg06IbaTsp9TFtqiuNhFTvmF8ZCDqWZNoubfnOah8gAvLMKOeNMctC5tVYZ/3r82LxPimKUFXjQFStvD8GKDNJJN8qiu0XRcVZAYAvbRKqkRVDaDBHkSBtDwYHkfmHHTze3JXk3LjiJp" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=3\\cdot\\:4\\left(\\sec\\left(θ\\right)\\right)^{3}\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$3\\cdot\\:4\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "3\\cdot\\:4\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$3\\cdot\\:4=12$$", "result": "=12\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{3+1}\\left(θ\\right)$$" ], "result": "=12\\sec^{3+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$3+1=4$$", "result": "=12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7jFGb0q9vd+kdWvVYeHJHbDNSoP5XQt+n/5y371e8QHJROlTYN926lBwmiJdD5HUSzRqDxPUzBN6vjj5oJL9kUFPzsE52Mw7P00gxEE61byKp9ej8WEAztP3UNEC9n7jjZEt3ZXAiqUE0HIXrrrezJHZKl2mV4AlztiApTAFigz2gNkWvAT/c3O3uDO/qxcKwM1Kg/ldC36f/nLfvV7xActytFlzCV8oq55bHLwXbb0c=" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=4\\left(-4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(12\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)\\right)=12\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(12\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=12\\frac{d}{dθ}\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=12\\left(\\frac{d}{dθ}\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)+\\frac{d}{dθ}\\left(\\sec^{6}\\left(θ\\right)\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)=4\\left(4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=4\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the Product Rule: $$\\left(f{\\cdot}g\\right)'=f'{\\cdot}g+f{\\cdot}g'$$", "secondary": [ "$$f=\\sec^{4}\\left(θ\\right),\\:g=\\tan^{2}\\left(θ\\right)$$" ], "result": "=4\\left(\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)\\tan^{2}\\left(θ\\right)+\\frac{d}{dθ}\\left(\\tan^{2}\\left(θ\\right)\\right)\\sec^{4}\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Product%20Rule", "practiceTopic": "Product Rule" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)=4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{4},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{4}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{4}\\right)=4u^{3}$$", "input": "\\frac{d}{du}\\left(u^{4}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=4u^{4-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=4u^{3}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYl3oEe4EC6n4B8Xgz0cKnkqk3hxk9aCfAWodBRxXgUexuzo0Fl6VhneutEteidn+Kv8//6/nV5O4fb8Xgwi7mapNFxUvwBeni+JEIFAdbegzYEOcQgyW4n51cR7AeMAm7Q==" } }, { "type": "step", "result": "=4u^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgA2CO5cm49xNEqFV0DMaZJRqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidVduWEnoDYDg06IbaTsp9TFtqiuNhFTvmF8ZCDqWZNoubfnOah8gAvLMKOeNMctC5tVYZ/3r82LxPimKUFXjQFStvD8GKDNJJN8qiu0XRcVZAYAvbRKqkRVDaDBHkSBtDwYHkfmHHTze3JXk3LjiJp" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$4\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "4\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{3+1}\\left(θ\\right)$$" ], "result": "=4\\sec^{3+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$3+1=4$$", "result": "=4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7p9w76+/nvHitDrMIupHrrrPmAPxuc/RFHWXkB8A+mnqLeJKmeakO1+Q3aSIldG41PCyA7ixjCdfDYe+yac7KOvOIrkYdKW3Q58EnXaMU+7oslHPSvPf5UfEs5T+IFZ4XHjb2+5NLFZrsH9fcPWg/TXL71sHPAsAQtYZvjpaGSRwboK/Igc1IRC3EGruCoRzV1Zrc/SYRIR8JuxYpD+PJ+Q==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan^{2}\\left(θ\\right)\\right)=2\\tan\\left(θ\\right)\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan^{2}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}2\\tan\\left(θ\\right)\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan^{2}\\left(θ\\right)\\right)", "result": "=2\\tan\\left(θ\\right)\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{2},\\:\\:u=\\tan\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{2}\\right)\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{2}\\right)=2u$$", "input": "\\frac{d}{du}\\left(u^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2u^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2u", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYr+VZUwBnLdzbS6DZQ+f4s+k3hxk9aCfAWodBRxXgUexMchyqTAJWrzJaDbnNcFsJUeCBKuYKgaNJ253gLI69U79qbCA2QqVmvm3jGRXZ2ppvbGT4j1utMEkCDH25m/vlQ==" } }, { "type": "step", "result": "=2u\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\tan\\left(θ\\right)$$", "result": "=2\\tan\\left(θ\\right)\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYux7TmDJ3fopnHGBxoqO+paSXIThcWH58oYoMPt18lzFqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidIn+HfeW3QQwAkRa15J0NSAlj501WLIv46IdbBaParYow2V0WdtbiweShXv1KOwqCjeh7+jKEzLb7VNCEMF3Z/Maswa5oIagKQFOfrnKm5+Gbb/PxX3awut5KxRxyZf8A=" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "result": "=\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYs0Vsx3c0nt53Utu5DgHTe3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK00HV/bLwwglXz/2YNx2i4JHggSrmCoGjSdud4CyOvVOgdQN0k+N3NHNAkEIoLrhB631k80nn+ODwI4ID0kCF5+z7zRz81SqRD02HEUE4Wqh" } }, { "type": "step", "result": "=2\\tan\\left(θ\\right)\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=4\\left(4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{2}\\left(θ\\right)+2\\tan\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Simplify $$4\\left(4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{2}\\left(θ\\right)+2\\tan\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right):{\\quad}4\\left(4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)$$", "input": "4\\left(4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{2}\\left(θ\\right)+2\\tan\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right)", "result": "=4\\left(4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "$$4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{2}\\left(θ\\right)=4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)$$", "input": "4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\tan\\left(θ\\right)\\tan^{2}\\left(θ\\right)=\\:\\tan^{1+2}\\left(θ\\right)$$" ], "result": "=4\\sec^{4}\\left(θ\\right)\\tan^{1+2}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+2=3$$", "result": "=4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7GHerUg4zB8o0EeBO6BKV162cmBc6Js8szDUhC2Hxdqi5HNbiQL8+p2P+VJTsJbU7q47vuWedXv2WUg94ER8IwVADhNIEz39BpXewdhMCBb+bbDkw/o6UIkLjzolgTAsqkuHx4ulZqfxZHIsiQUpV1nyo9elyUFFpcPZW02cAt7EtbJdlgVP3iSyKFg6v1zm6cUwY1HWd9EJWOCb4O+kauMdM6yd/xAZrmYAkuRY8/rYkt3WiGR7ZaCaXvz77bMjS" } }, { "type": "interim", "title": "$$2\\tan\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)=2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "2\\tan\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)=\\:\\sec^{2+4}\\left(θ\\right)$$" ], "result": "=2\\tan\\left(θ\\right)\\sec^{2+4}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+4=6$$", "result": "=2\\tan\\left(θ\\right)\\sec^{6}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7OyotidIxGYYp2AND0rbhNaCGttAzkBWvgG18MKDLv3Rdbcltti6PzkTDSwkmBTY6q47vuWedXv2WUg94ER8IwSIYCdSNn6m2gNv8FkjgS1XBL9WNCz07kx2T4iGijNtQjT/+cTnIKvMqhzKaFBbxvzSei4QZxxrazp/cx7cYZvbVeYdTw2sAeZ9o0nz1uUihamteA8UyY/eLm9jagP57hkwLhsnE4D1C6IGVVz/pqlY=" } }, { "type": "step", "result": "=4\\left(2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)+4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7iza9NvP7nq2MIMeLHhROsvkdSragnZ5K8+WMRM5qjJ4/XhyePE1SthuGDJcaFMnncKXVLLQXMot2gtnZH7qY51d1u0XlM+NdfDNBhSESuvlV00rpv8+ZC6TM10tVCSHsi/RlR31DZv0142/jAcann2ASrhThz5UQhS79xvpIR8tIoCq5NivR6sCY2Sfa33kc8jzDW/n9HorFBpgomiqIe+5byrQDQVCXUD0vH/fvOdxyhd7tjiG+GxQNxDvGkZUlqqyy15Nuz77zqSegKeIT2iKI9Km/ZvzCnLF9eqmklKHqRJDJ/DeTzEayKJpmRlyWQi730E1b9JEo9956Qy3sPLLWww1IW5GUOlAGrcUL2OqwiNrEngO+NNvZ9sqNu+2V" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec^{6}\\left(θ\\right)\\right)=6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{6}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}6\\left(\\sec\\left(θ\\right)\\right)^{5}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{6}\\left(θ\\right)\\right)", "result": "=6\\left(\\sec\\left(θ\\right)\\right)^{5}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{6},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{6}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{6}\\right)=6u^{5}$$", "input": "\\frac{d}{du}\\left(u^{6}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=6u^{6-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=6u^{5}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYgOK76JOSNqGa1B1wU6ozWWk3hxk9aCfAWodBRxXgUexqSoDkuZ4Ab1St3bM8oqKcP8//6/nV5O4fb8Xgwi7mapNFxUvwBeni+JEIFAdbegzHfwjUTnSu4cWqw+8m79+NA==" } }, { "type": "step", "result": "=6u^{5}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=6\\left(\\sec\\left(θ\\right)\\right)^{5}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgBaJTGF5uPwSMzxoSyJNVZsqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKieMSe35zYqdvyntnEYxaQ88jJq7JZFtKgvxzzzuaqTGrebfnOah8gAvLMKOeNMctC5tVYZ/3r82LxPimKUFXjQFStvD8GKDNJJN8qiu0XRcVZAYAvbRKqkRVDaDBHkSBtDwYHkfmHHTze3JXk3LjiJp" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=6\\left(\\sec\\left(θ\\right)\\right)^{5}\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$6\\sec^{5}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "6\\sec^{5}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{5}\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{5+1}\\left(θ\\right)$$" ], "result": "=6\\sec^{5+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$5+1=6$$", "result": "=6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7QMJmgA2Sx2hYpGS5j/h1NLPmAPxuc/RFHWXkB8A+mnqLeJKmeakO1+Q3aSIldG417JiujyaxykLR7Tv0Nr+wOGprXgPFMmP3i5vY2oD+e4YslHPSvPf5UfEs5T+IFZ4XHjb2+5NLFZrsH9fcPWg/TRiIlnZT/wCWlhB0uSVNtvx/uR/EZqc6EbI3tnxSI5001Zrc/SYRIR8JuxYpD+PJ+Q==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=12\\left(4\\left(4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Expand $$4\\left(4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "4\\left(4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=12\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Expand $$4\\left(4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right):{\\quad}16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+8\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "4\\left(4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)", "result": "=16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+8\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)+6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b+c\\right)=ab+ac$$", "secondary": [ "$$a=4,\\:b=4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right),\\:c=2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)$$" ], "result": "=4\\cdot\\:4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+4\\cdot\\:2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "interim", "title": "Simplify $$4\\cdot\\:4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+4\\cdot\\:2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+8\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "4\\cdot\\:4\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+4\\cdot\\:2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+8\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:4=16$$", "result": "=16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+4\\cdot\\:2\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:2=8$$", "result": "=16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+8\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7iza9NvP7nq2MIMeLHhROsoL8UIMToMV6hDKJhY+Nk7toxrHhSJArgBpsnuAV9UCfBgOJ+lz5D5t+hsCNYCoDMHWD310L1+P2yDQQfMEhENFHx6CvpVa90MvMzklpzJWXZAzoThz1g+TITtf+WJ7KiyRRh7kCo0obc/CTaNyPGkYuuAKnMbBNnPJ2x/QM7ydV1sD7NfhsPe7eDHrmjY0mEyzoLVQPyp9R+5GC0prpwlf2K8DiYaJsjNsjOcdOcdQLW6BryIKCtddVLZsZyZ1dV9DmtpfPTl/o4ZJKfQIdsQqwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "primary": "Add similar elements: $$8\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)+6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)=14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7iza9NvP7nq2MIMeLHhROsoL8UIMToMV6hDKJhY+Nk7toxrHhSJArgBpsnuAV9UCf8SVGtBi1ovwgT5vDB2QfkF3B8WOxYrIgIdHvusvufLAAlilG71elit3w1IBbYN0PumuR+HimzGDTYbsfY1Q6JXyo9elyUFFpcPZW02cAt7EXYiLGBL7PG2xAqJd1w6BXKK1Q1tVX7SUdYWONKsZTX66q9beEAcfrflX+8QRghq2CcvL2jSzw1VlRcGX4ka6qwZ5KNwm43WeyBMBBKL30QvI8w1v5/R6KxQaYKJoqiHsLiDrnJdtlx1I2mhkkUTElN8b/k79QNO9t6sAoPBR3zJYojigZ8i1ymB3KwWBswnjiAEmXhYw7WsDRrfT9tRiW" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=8\\left(-4\\left(-4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+12\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(24\\left(64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+132\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+14\\sec^{8}\\left(θ\\right)\\right)\\right)=24\\left(256\\tan^{5}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+1048\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+376\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(24\\left(64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+132\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+14\\sec^{8}\\left(θ\\right)\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=24\\frac{d}{dθ}\\left(64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+132\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+14\\sec^{8}\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=24\\left(\\frac{d}{dθ}\\left(64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)\\right)+\\frac{d}{dθ}\\left(132\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)+\\frac{d}{dθ}\\left(14\\sec^{8}\\left(θ\\right)\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)\\right)=64\\left(4\\tan^{5}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+4\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=64\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the Product Rule: $$\\left(f{\\cdot}g\\right)'=f'{\\cdot}g+f{\\cdot}g'$$", "secondary": [ "$$f=\\sec^{4}\\left(θ\\right),\\:g=\\tan^{4}\\left(θ\\right)$$" ], "result": "=64\\left(\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)\\tan^{4}\\left(θ\\right)+\\frac{d}{dθ}\\left(\\tan^{4}\\left(θ\\right)\\right)\\sec^{4}\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Product%20Rule", "practiceTopic": "Product Rule" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)=4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{4},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{4}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{4}\\right)=4u^{3}$$", "input": "\\frac{d}{du}\\left(u^{4}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=4u^{4-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=4u^{3}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYl3oEe4EC6n4B8Xgz0cKnkqk3hxk9aCfAWodBRxXgUexuzo0Fl6VhneutEteidn+Kv8//6/nV5O4fb8Xgwi7mapNFxUvwBeni+JEIFAdbegzYEOcQgyW4n51cR7AeMAm7Q==" } }, { "type": "step", "result": "=4u^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgA2CO5cm49xNEqFV0DMaZJRqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidVduWEnoDYDg06IbaTsp9TFtqiuNhFTvmF8ZCDqWZNoubfnOah8gAvLMKOeNMctC5tVYZ/3r82LxPimKUFXjQFStvD8GKDNJJN8qiu0XRcVZAYAvbRKqkRVDaDBHkSBtDwYHkfmHHTze3JXk3LjiJp" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$4\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "4\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{3+1}\\left(θ\\right)$$" ], "result": "=4\\sec^{3+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$3+1=4$$", "result": "=4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7p9w76+/nvHitDrMIupHrrrPmAPxuc/RFHWXkB8A+mnqLeJKmeakO1+Q3aSIldG41PCyA7ixjCdfDYe+yac7KOvOIrkYdKW3Q58EnXaMU+7oslHPSvPf5UfEs5T+IFZ4XHjb2+5NLFZrsH9fcPWg/TXL71sHPAsAQtYZvjpaGSRwboK/Igc1IRC3EGruCoRzV1Zrc/SYRIR8JuxYpD+PJ+Q==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan^{4}\\left(θ\\right)\\right)=4\\tan^{3}\\left(θ\\right)\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan^{4}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}4\\left(\\tan\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan^{4}\\left(θ\\right)\\right)", "result": "=4\\left(\\tan\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{4},\\:\\:u=\\tan\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{4}\\right)\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{4}\\right)=4u^{3}$$", "input": "\\frac{d}{du}\\left(u^{4}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=4u^{4-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=4u^{3}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYl3oEe4EC6n4B8Xgz0cKnkqk3hxk9aCfAWodBRxXgUexuzo0Fl6VhneutEteidn+Kv8//6/nV5O4fb8Xgwi7mapNFxUvwBeni+JEIFAdbegzYEOcQgyW4n51cR7AeMAm7Q==" } }, { "type": "step", "result": "=4u^{3}\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\tan\\left(θ\\right)$$", "result": "=4\\left(\\tan\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYux7TmDJ3fopnHGBxoqO+pY2CO5cm49xNEqFV0DMaZJRqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidVduWEnoDYDg06IbaTsp9TzW39yg3f07n4Kh0WH31pB8q1KfCFKwSwA0qmWovUVP9tVYZ/3r82LxPimKUFXjQFStvD8GKDNJJN8qiu0XRcVZAYAvbRKqkRVDaDBHkSBtDwYHkfmHHTze3JXk3LjiJp" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "result": "=\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYs0Vsx3c0nt53Utu5DgHTe3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK00HV/bLwwglXz/2YNx2i4JHggSrmCoGjSdud4CyOvVOgdQN0k+N3NHNAkEIoLrhB631k80nn+ODwI4ID0kCF5+z7zRz81SqRD02HEUE4Wqh" } }, { "type": "step", "result": "=4\\left(\\tan\\left(θ\\right)\\right)^{3}\\sec^{2}\\left(θ\\right)" }, { "type": "step", "primary": "Simplify", "result": "=4\\tan^{3}\\left(θ\\right)\\sec^{2}\\left(θ\\right)", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=64\\left(4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{4}\\left(θ\\right)+4\\tan^{3}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Simplify $$64\\left(4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{4}\\left(θ\\right)+4\\tan^{3}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right):{\\quad}64\\left(4\\tan^{5}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+4\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)\\right)$$", "input": "64\\left(4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{4}\\left(θ\\right)+4\\tan^{3}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right)", "result": "=64\\left(4\\tan^{5}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+4\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "$$4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{4}\\left(θ\\right)=4\\tan^{5}\\left(θ\\right)\\sec^{4}\\left(θ\\right)$$", "input": "4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{4}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\tan\\left(θ\\right)\\tan^{4}\\left(θ\\right)=\\:\\tan^{1+4}\\left(θ\\right)$$" ], "result": "=4\\sec^{4}\\left(θ\\right)\\tan^{1+4}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+4=5$$", "result": "=4\\sec^{4}\\left(θ\\right)\\tan^{5}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7GHerUg4zB8o0EeBO6BKV162cmBc6Js8szDUhC2Hxdqhdbcltti6PzkTDSwkmBTY6q47vuWedXv2WUg94ER8IwVADhNIEz39BpXewdhMCBb/U7dZxYBZ6VQ59yTO/BTRekuHx4ulZqfxZHIsiQUpV1nyo9elyUFFpcPZW02cAt7Hms8g3dTKBs8Qcved1uTbvGp3//8xunxxjTjNOU2Uj8WAEODKzAB5RvbiyYeFhfmIkt3WiGR7ZaCaXvz77bMjS" } }, { "type": "interim", "title": "$$4\\tan^{3}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)=4\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)$$", "input": "4\\tan^{3}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)=\\:\\sec^{2+4}\\left(θ\\right)$$" ], "result": "=4\\tan^{3}\\left(θ\\right)\\sec^{2+4}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+4=6$$", "result": "=4\\tan^{3}\\left(θ\\right)\\sec^{6}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Q5Qt/9UZA+6+eHizBC+zZOH0qCTik4j9uS7jTF7ng2/8DCHl2/1cUFH2kQZvGLuVzRqDxPUzBN6vjj5oJL9kUH0lfPvVpT4vMUEVz1kntu7WxcLMxZwB9cGBLGbwbyjTrqr1t4QBx+t+Vf7xBGCGrcGZ/v+w0kROeeUMRN/Dzfvh9Kgk4pOI/bku40xe54NvceVwY//SmIC3EDK+C+Nj9Yx6/teVLREoXphSPTLZ2MaMe0O2wYArgGZd6UMSRoCP" } }, { "type": "step", "result": "=64\\left(4\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+4\\tan^{5}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7A5AfjU1SRWvT8rmMMzVgcBiMJmjxigW8LortZKvrqEZDn/Xi9tAk0BagOH0ipjvtP2VpgCWDQiLGOtegj6K/y/padcGORE5bHX37Io3viWolp/MB1OadflBmLQoq5Pmlo5FYteSPKwXny4uCMrdsK87yytEh92Rj5rzFYDrjNFetlKq927IFCE0mSgRkqm2pGWXl+3rXhjFFYJ9jApz9IbNYmw5UnCphczNkGo2TpwyBBTEk/JQ2cZ9WKuRzClU7VkZbu12YlUPSk9mOI6WLjyHQ+8WidKe5mLtnoi0iGZg+SAm6ZAjmyYLg+gLSGx4mXr7wSwGQCg6Xu6w7Iszv6TSei4QZxxrazp/cx7cYZvYA0hWLB/nt1m4ggrh+mxlbvzIPeEtDfcHv/z8uls8Teg==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(132\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)=132\\left(6\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+2\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(132\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=132\\frac{d}{dθ}\\left(\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the Product Rule: $$\\left(f{\\cdot}g\\right)'=f'{\\cdot}g+f{\\cdot}g'$$", "secondary": [ "$$f=\\sec^{6}\\left(θ\\right),\\:g=\\tan^{2}\\left(θ\\right)$$" ], "result": "=132\\left(\\frac{d}{dθ}\\left(\\sec^{6}\\left(θ\\right)\\right)\\tan^{2}\\left(θ\\right)+\\frac{d}{dθ}\\left(\\tan^{2}\\left(θ\\right)\\right)\\sec^{6}\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Product%20Rule", "practiceTopic": "Product Rule" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec^{6}\\left(θ\\right)\\right)=6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{6}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}6\\left(\\sec\\left(θ\\right)\\right)^{5}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{6}\\left(θ\\right)\\right)", "result": "=6\\left(\\sec\\left(θ\\right)\\right)^{5}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{6},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{6}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{6}\\right)=6u^{5}$$", "input": "\\frac{d}{du}\\left(u^{6}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=6u^{6-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=6u^{5}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYgOK76JOSNqGa1B1wU6ozWWk3hxk9aCfAWodBRxXgUexqSoDkuZ4Ab1St3bM8oqKcP8//6/nV5O4fb8Xgwi7mapNFxUvwBeni+JEIFAdbegzHfwjUTnSu4cWqw+8m79+NA==" } }, { "type": "step", "result": "=6u^{5}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=6\\left(\\sec\\left(θ\\right)\\right)^{5}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgBaJTGF5uPwSMzxoSyJNVZsqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKieMSe35zYqdvyntnEYxaQ88jJq7JZFtKgvxzzzuaqTGrebfnOah8gAvLMKOeNMctC5tVYZ/3r82LxPimKUFXjQFStvD8GKDNJJN8qiu0XRcVZAYAvbRKqkRVDaDBHkSBtDwYHkfmHHTze3JXk3LjiJp" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=6\\left(\\sec\\left(θ\\right)\\right)^{5}\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$6\\sec^{5}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "6\\sec^{5}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{5}\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{5+1}\\left(θ\\right)$$" ], "result": "=6\\sec^{5+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$5+1=6$$", "result": "=6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7QMJmgA2Sx2hYpGS5j/h1NLPmAPxuc/RFHWXkB8A+mnqLeJKmeakO1+Q3aSIldG417JiujyaxykLR7Tv0Nr+wOGprXgPFMmP3i5vY2oD+e4YslHPSvPf5UfEs5T+IFZ4XHjb2+5NLFZrsH9fcPWg/TRiIlnZT/wCWlhB0uSVNtvx/uR/EZqc6EbI3tnxSI5001Zrc/SYRIR8JuxYpD+PJ+Q==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan^{2}\\left(θ\\right)\\right)=2\\tan\\left(θ\\right)\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan^{2}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}2\\tan\\left(θ\\right)\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan^{2}\\left(θ\\right)\\right)", "result": "=2\\tan\\left(θ\\right)\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{2},\\:\\:u=\\tan\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{2}\\right)\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{2}\\right)=2u$$", "input": "\\frac{d}{du}\\left(u^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2u^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2u", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYr+VZUwBnLdzbS6DZQ+f4s+k3hxk9aCfAWodBRxXgUexMchyqTAJWrzJaDbnNcFsJUeCBKuYKgaNJ253gLI69U79qbCA2QqVmvm3jGRXZ2ppvbGT4j1utMEkCDH25m/vlQ==" } }, { "type": "step", "result": "=2u\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\tan\\left(θ\\right)$$", "result": "=2\\tan\\left(θ\\right)\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYux7TmDJ3fopnHGBxoqO+paSXIThcWH58oYoMPt18lzFqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidIn+HfeW3QQwAkRa15J0NSAlj501WLIv46IdbBaParYow2V0WdtbiweShXv1KOwqCjeh7+jKEzLb7VNCEMF3Z/Maswa5oIagKQFOfrnKm5+Gbb/PxX3awut5KxRxyZf8A=" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "result": "=\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYs0Vsx3c0nt53Utu5DgHTe3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK00HV/bLwwglXz/2YNx2i4JHggSrmCoGjSdud4CyOvVOgdQN0k+N3NHNAkEIoLrhB631k80nn+ODwI4ID0kCF5+z7zRz81SqRD02HEUE4Wqh" } }, { "type": "step", "result": "=2\\tan\\left(θ\\right)\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=132\\left(6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{2}\\left(θ\\right)+2\\tan\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{6}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Simplify $$132\\left(6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{2}\\left(θ\\right)+2\\tan\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{6}\\left(θ\\right)\\right):{\\quad}132\\left(6\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+2\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)\\right)$$", "input": "132\\left(6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{2}\\left(θ\\right)+2\\tan\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{6}\\left(θ\\right)\\right)", "result": "=132\\left(6\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+2\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "$$6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{2}\\left(θ\\right)=6\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)$$", "input": "6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\tan\\left(θ\\right)\\tan^{2}\\left(θ\\right)=\\:\\tan^{1+2}\\left(θ\\right)$$" ], "result": "=6\\sec^{6}\\left(θ\\right)\\tan^{1+2}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+2=3$$", "result": "=6\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7H4jwT8QhunkiDAGNb+Rcva2cmBc6Js8szDUhC2Hxdqi5HNbiQL8+p2P+VJTsJbU7q47vuWedXv2WUg94ER8IwZGREjArFZVzoA4Cy52193mbbDkw/o6UIkLjzolgTAsqwqP97mnHf8zpjilA0FhQHuj4b/xSLwXZQVzICbe8+7UtbJdlgVP3iSyKFg6v1zm6XXbfFRrmBUC90vtWO60GksdM6yd/xAZrmYAkuRY8/rYkt3WiGR7ZaCaXvz77bMjS" } }, { "type": "interim", "title": "$$2\\tan\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{6}\\left(θ\\right)=2\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "2\\tan\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{6}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{2}\\left(θ\\right)\\sec^{6}\\left(θ\\right)=\\:\\sec^{2+6}\\left(θ\\right)$$" ], "result": "=2\\tan\\left(θ\\right)\\sec^{2+6}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+6=8$$", "result": "=2\\tan\\left(θ\\right)\\sec^{8}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7OyotidIxGYYp2AND0rbhNaCGttAzkBWvgG18MKDLv3TbMUomGEDalcbJrsz1ZKu/q47vuWedXv2WUg94ER8IwSIYCdSNn6m2gNv8FkjgS1Wta2mu2dy0k1Wp64KXS/1rjT/+cTnIKvMqhzKaFBbxvzSei4QZxxrazp/cx7cYZvb1XqCxOosrrwiv/aS1ROCGSHdJEyJPexqhQEbR26e/s0wLhsnE4D1C6IGVVz/pqlY=" } }, { "type": "step", "result": "=132\\left(2\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)+6\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7PV+l92IXFa8mTDEwIfC7DCKI9Km/ZvzCnLF9eqmklKHqRJDJ/DeTzEayKJpmRlyWQi730E1b9JEo9956Qy3sPJgndhfIfsC9I7wDMrBBRj8AlilG71elit3w1IBbYN0PB5Bs5lVGb45zGNFiEvBODcjDyWbJQVk1pvf07Y462fJXV7Qqnb+eOc6LPNN3xs8nyZ0rztQs0QA5MTJACM+fadWc8BVevEVI1mgOPLFPH8CLGmNnLPWGf9PH3lpmjoJIZIRaf9fTURBjTQ6UkTGlImtPcLmP23q51gN6YJjM09cr8Xmq4XLWwD8V90BgYeZJP2VXWb3eVFnN+G/7GJIl8HXdshbMhQZCHpp35utrS0G0Y1gka5PyQUgt+7WzOOr4" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(14\\sec^{8}\\left(θ\\right)\\right)=112\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(14\\sec^{8}\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=14\\frac{d}{dθ}\\left(\\sec^{8}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Apply the chain rule:$${\\quad}8\\left(\\sec\\left(θ\\right)\\right)^{7}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{8}\\left(θ\\right)\\right)", "result": "=8\\left(\\sec\\left(θ\\right)\\right)^{7}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{8},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{8}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{8}\\right)=8u^{7}$$", "input": "\\frac{d}{du}\\left(u^{8}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=8u^{8-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=8u^{7}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqfCu3R3nHIhDvsPcLLz/1Kk3hxk9aCfAWodBRxXgUexxBsJp2NepDuy/SDgSGtTQP8//6/nV5O4fb8Xgwi7mapNFxUvwBeni+JEIFAdbegz9Cf5/xFJQkGUmQmSfOtWkA==" } }, { "type": "step", "result": "=8u^{7}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=8\\left(\\sec\\left(θ\\right)\\right)^{7}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgAmdV6hJ9cldZkSASENQvDKqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKie/FI4A8N6pVJJZl5fBw9jL2Bb9Z084y+BADz+fW/X+8ObfnOah8gAvLMKOeNMctC5tVYZ/3r82LxPimKUFXjQFStvD8GKDNJJN8qiu0XRcVZAYAvbRKqkRVDaDBHkSBtDwYHkfmHHTze3JXk3LjiJp" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=14\\cdot\\:8\\left(\\sec\\left(θ\\right)\\right)^{7}\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$14\\cdot\\:8\\sec^{7}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}112\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "14\\cdot\\:8\\sec^{7}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=112\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$14\\cdot\\:8=112$$", "result": "=112\\sec^{7}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{7}\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{7+1}\\left(θ\\right)$$" ], "result": "=112\\sec^{7+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$7+1=8$$", "result": "=112\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7qqpiC4le3DAJBVqCch5PKwVLcfLCp9Boesbd0nXQesBqKrdqZr3xi0oZ0kf4/81Y3XeO2tIUPH5Q2xrCOU6NXawGQ5eg9xcOzW/RgEfPor9DSTWuNGAj+wIwMGsd7LCW7lvKtANBUJdQPS8f9+853HKF3u2OIb4bFA3EO8aRlSVMbZv4Wa4I2LaPOrVXIZtj9A5Bkci853Hfrrw5WP/odGZiNlnz9jzyUG5gNa45hnA=" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=24\\left(64\\left(4\\tan^{5}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+4\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)\\right)+132\\left(6\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+2\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+112\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Expand $$64\\left(4\\tan^{5}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+4\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)\\right)+132\\left(6\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+2\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+112\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}256\\tan^{5}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+1048\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+376\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "64\\left(4\\tan^{5}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+4\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)\\right)+132\\left(6\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+2\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+112\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=24\\left(256\\tan^{5}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+1048\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+376\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Expand $$64\\left(4\\tan^{5}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+4\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)\\right):{\\quad}256\\tan^{5}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+256\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)$$", "input": "64\\left(4\\tan^{5}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+4\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)\\right)", "result": "=256\\tan^{5}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+256\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+132\\left(6\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+2\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+112\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b+c\\right)=ab+ac$$", "secondary": [ "$$a=64,\\:b=4\\tan^{5}\\left(θ\\right)\\sec^{4}\\left(θ\\right),\\:c=4\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)$$" ], "result": "=64\\cdot\\:4\\tan^{5}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+64\\cdot\\:4\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "primary": "Multiply the numbers: $$64\\cdot\\:4=256$$", "result": "=256\\tan^{5}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+256\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)" } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7YkvlWLayjKN6FXarLQ0GJcTHt7EfGTKANmJ5adlvOd46a6whJcHEYdFzx1pH94gcENAXTRYlkeXi/6iJJcYG3N6GQqufR6tr2vPxOUv7H+8zdEfU9seBjOhaIUijVXXbLyPYwflVgHzSz1oAHVz6pos4Mes11HpSwidsI6oRyK63hVfpOpUStEwbilPE4ogd+V7S6kFzr6zNGXs78LxaihJyf8zawtgEaDEKWrMLEzb8ak59hLGKbUsPp4qZlO8pI0RxNAAeCiAupyTRxbXafXxAop4KFNvJdALgGVyiwSDyPMNb+f0eisUGmCiaKoh74gBJl4WMO1rA0a30/bUYlg==" } }, { "type": "interim", "title": "Expand $$132\\left(6\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+2\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)\\right):{\\quad}792\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+264\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "132\\left(6\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+2\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)\\right)", "result": "=256\\tan^{5}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+256\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+792\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+264\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)+112\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b+c\\right)=ab+ac$$", "secondary": [ "$$a=132,\\:b=6\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right),\\:c=2\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)$$" ], "result": "=132\\cdot\\:6\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+132\\cdot\\:2\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "interim", "title": "Simplify $$132\\cdot\\:6\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+132\\cdot\\:2\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}792\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+264\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "132\\cdot\\:6\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+132\\cdot\\:2\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=792\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+264\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$132\\cdot\\:6=792$$", "result": "=792\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+132\\cdot\\:2\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "step", "primary": "Multiply the numbers: $$132\\cdot\\:2=264$$", "result": "=792\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+264\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7PV+l92IXFa8mTDEwIfC7DGQM6E4c9YPkyE7X/lieyos/6pCB86paeQ1o/mNgPa+jsFDSg/SgXmHLlF40pckVgc0ag8T1MwTer44+aCS/ZFArZHiZ7NuNPnvM2r6E9STNyZ0rztQs0QA5MTJACM+faQTXQmcLJlJVdaT/WTcsWRj8WaGp/4qrIjQsifAUsfjIo3oe/oyhMy2+1TQhDBd2f3FGtZiGdBdn/vMAKrUamk+2lifHQThNv1usau5svJJWtITS+u4dgApxM44qzMwAP0h3SRMiT3saoUBG0dunv7Ov8EPIWSoKCGyUa9XWGgj9" } }, { "type": "interim", "title": "Simplify $$256\\tan^{5}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+256\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+792\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+264\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)+112\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}256\\tan^{5}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+1048\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+376\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "256\\tan^{5}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+256\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+792\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+264\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)+112\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=256\\tan^{5}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+1048\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+376\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Add similar elements: $$256\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+792\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)=1048\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)$$", "result": "=256\\tan^{5}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+1048\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+264\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)+112\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "step", "primary": "Add similar elements: $$264\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)+112\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)=376\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=256\\tan^{5}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+1048\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+376\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7YkvlWLayjKN6FXarLQ0GJcTHt7EfGTKANmJ5adlvOd46a6whJcHEYdFzx1pH94gcwFfeXnM/lvx01JBoc1o6AsBktQ+t8hyUFoIA4Yi8yCc1irG62HmwF+IYGSOyqLSzXCsnbF38Nu9G1Z+Aq4NC5V+3yJQHRqfhEVf8c5U3TYfCLrvQLCwPdPD4IML/t1EYAJYpRu9XpYrd8NSAW2DdD5m8zKeFKVa8zuIV0MYCHy4ZZeX7eteGMUVgn2MCnP0hdyRR/hNAoQD2hud6UQuWgOj4b/xSLwXZQVzICbe8+7UyTWkzTJLrtTWPrVqAX2AVSdIld1bs2y/sWqCCCM0iyrqcSX+26Db9UvR3NA1yV77e1A9ekahUIWtvNWrN4v94TqjrSsmR8+NC639wiDwOlPpadcGORE5bHX37Io3viWqUOWIbI0Do3dV5cylo4HgpU4MWJVmSI5n5HboYO2WOTs3oTzs9Co/R/sW57aWygHyC/FCDE6DFeoQyiYWPjZO7JbW2WbaO0FHShlZF9YPATz0ppTTFwEo27GgTvRxf9XuMhtcBGNifOMBvB0yOtQ2TVNn4NhKPJIaEH5yvz96HJA==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=-8\\left(-4\\left(-4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+12\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)\\right)+24\\left(256\\tan^{5}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+1048\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+376\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{d}{dθ}\\left(-8\\left(-4\\left(-4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+12\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)\\right)+24\\left(256\\tan^{5}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+1048\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+376\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(-8\\left(-4\\left(-4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+12\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)\\right)+24\\left(256\\tan^{5}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+1048\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+376\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)\\right)\\right)=-8\\left(-4\\left(-4\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)+12\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)\\right)+12\\left(64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+132\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+14\\sec^{8}\\left(θ\\right)\\right)\\right)+24\\left(7568\\sec^{6}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+1024\\tan^{6}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+6152\\sec^{8}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+376\\sec^{10}\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(-8\\left(-4\\left(-4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+12\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)\\right)+24\\left(256\\tan^{5}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+1048\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+376\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=-\\frac{d}{dθ}\\left(8\\left(-4\\left(-4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+12\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)\\right)\\right)+\\frac{d}{dθ}\\left(24\\left(256\\tan^{5}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+1048\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+376\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(8\\left(-4\\left(-4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+12\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)\\right)\\right)=8\\left(-4\\left(-4\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)+12\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)\\right)+12\\left(64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+132\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+14\\sec^{8}\\left(θ\\right)\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(8\\left(-4\\left(-4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+12\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=8\\frac{d}{dθ}\\left(-4\\left(-4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+12\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)\\right)" }, { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=8\\left(-\\frac{d}{dθ}\\left(4\\left(-4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)\\right)+\\frac{d}{dθ}\\left(12\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(4\\left(-4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)\\right)=4\\left(-4\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)+12\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(4\\left(-4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=4\\frac{d}{dθ}\\left(-4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)+12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=4\\left(-\\frac{d}{dθ}\\left(4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\right)+\\frac{d}{dθ}\\left(12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\right)=4\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(4\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=4\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the Product Rule: $$\\left(f{\\cdot}g\\right)'=f'{\\cdot}g+f{\\cdot}g'$$", "secondary": [ "$$f=\\sec^{2}\\left(θ\\right),\\:g=\\tan\\left(θ\\right)$$" ], "result": "=4\\left(\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)\\tan\\left(θ\\right)+\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)\\sec^{2}\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Product%20Rule", "practiceTopic": "Product Rule" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{2}\\left(θ\\right)\\right)", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{2},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{2}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{2}\\right)=2u$$", "input": "\\frac{d}{du}\\left(u^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2u^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2u", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYr+VZUwBnLdzbS6DZQ+f4s+k3hxk9aCfAWodBRxXgUexMchyqTAJWrzJaDbnNcFsJUeCBKuYKgaNJ253gLI69U79qbCA2QqVmvm3jGRXZ2ppvbGT4j1utMEkCDH25m/vlQ==" } }, { "type": "step", "result": "=2u\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=2\\sec\\left(θ\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgCSXIThcWH58oYoMPt18lzFqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidZClirlWNFPnHR1rvdyoFLI7IC+oqnwb1T6nPLpiw1eysw2NXPd8jZnr3fcPpqc7ejeh7+jKEzLb7VNCEMF3Z/Maswa5oIagKQFOfrnKm5+Gbb/PxX3awut5KxRxyZf8A=" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "2\\sec\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{1+1}\\left(θ\\right)$$" ], "result": "=2\\sec^{1+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7aY4rYekMZhUfP6sZ6mfqd6Lsz0lt1XLEr88Kc/cjJo5V00rpv8+ZC6TM10tVCSHspqrCcTwb/I0f8OwMjuU6eBAUa+AHsHd5OGpMiuFITjOuqvW3hAHH635V/vEEYIatHimBRYRqHSWeJkuUPhfTC5LQ0B1JNgsb5SXzbS+SCnAfGB9H25Zc+Ua1z7Kw8kgb4gBJl4WMO1rA0a30/bUYlg==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "result": "=\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYs0Vsx3c0nt53Utu5DgHTe3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK00HV/bLwwglXz/2YNx2i4JHggSrmCoGjSdud4CyOvVOgdQN0k+N3NHNAkEIoLrhB631k80nn+ODwI4ID0kCF5+z7zRz81SqRD02HEUE4Wqh" } }, { "type": "step", "result": "=4\\left(2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Simplify $$4\\left(2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\right):{\\quad}4\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)$$", "input": "4\\left(2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\right)", "result": "=4\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "$$2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)=2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)$$", "input": "2\\sec^{2}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\tan\\left(θ\\right)\\tan\\left(θ\\right)=\\:\\tan^{1+1}\\left(θ\\right)$$" ], "result": "=2\\sec^{2}\\left(θ\\right)\\tan^{1+1}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Md1SJKHYLxhOYRRbsLTu+axUZgt6FTmFSHnRsotkqemLeJKmeakO1+Q3aSIldG41s7BwSVdKEYSqn+4fKAQoGk0xmFIl1KTO5aAkhaHvKzBALhShbMGZIZsZWTO721HsWI4dG63F/aSz/PFDvu5FnO3sngYNzcydIedGQjjtM8sBMUhkLMqDD6KHvgZAgKMdRQzIIgXxnWeDhDTk2Lfum8RPUSGriaFRn0NZXHZrPQw=" } }, { "type": "interim", "title": "$$\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)=\\sec^{4}\\left(θ\\right)$$", "input": "\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)=\\:\\sec^{2+2}\\left(θ\\right)$$" ], "result": "=\\sec^{2+2}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+2=4$$", "result": "=\\sec^{4}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7AtrsReczPdUqIdn5/Ev5mFtNfVJqszzpMwg6spHO8F8gJ/ZZA32ZInFBpDtxBfiKn3Jyj2VJgBojKQHjINpkkETYvnjqwhFu6aWmDOYpAVMO/U6qyRZonn1WPQ/21pz3cKXVLLQXMot2gtnZH7qY5zkbLioo7YVO0SRnDAO/7Uokt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "=4\\left(\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Simplify $$2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{4}\\left(θ\\right):{\\quad}3\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)$$", "input": "2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{4}\\left(θ\\right)", "result": "=4\\left(3\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Rewrite using trig identities", "input": "\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\tan^{2}\\left(θ\\right)", "result": "=-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Use the Pythagorean identity: $$\\tan^{2}\\left(x\\right)+1=\\sec^{2}\\left(x\\right)$$", "secondary": [ "$$\\tan^{2}\\left(x\\right)=\\sec^{2}\\left(x\\right)-1$$" ], "result": "=\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right)" }, { "type": "interim", "title": "Simplify $$\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right):{\\quad}3\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)$$", "input": "\\sec^{4}\\left(θ\\right)+2\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right)", "result": "=3\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "interim", "title": "Expand $$2\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right):{\\quad}2\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)$$", "input": "2\\sec^{2}\\left(θ\\right)\\left(\\sec^{2}\\left(θ\\right)-1\\right)", "result": "=\\sec^{4}\\left(θ\\right)+2\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b-c\\right)=ab-ac$$", "secondary": [ "$$a=2\\sec^{2}\\left(θ\\right),\\:b=\\sec^{2}\\left(θ\\right),\\:c=1$$" ], "result": "=2\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)\\cdot\\:1", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "result": "=2\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)-2\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$2\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)-2\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right):{\\quad}2\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)$$", "input": "2\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)-2\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right)", "result": "=2\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "interim", "title": "$$2\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)=2\\sec^{4}\\left(θ\\right)$$", "input": "2\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)=\\:\\sec^{2+2}\\left(θ\\right)$$" ], "result": "=2\\sec^{2+2}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+2=4$$", "result": "=2\\sec^{4}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Md1SJKHYLxhOYRRbsLTu+dJYPDPDesonRI520G8zYRPehkKrn0era9rz8TlL+x/vyvn14PieOyuPP98CkiPIaXRfFgXB+vlRb/AHCJklpF+qQC22PXYzQL8KzAhJCzszqX/cSuhK5Ty2xqd/IdNr67A2LXxYB8f3C3KpWaLgll1U2fg2Eo8khoQfnK/P3ock" } }, { "type": "interim", "title": "$$2\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right)=2\\sec^{2}\\left(θ\\right)$$", "input": "2\\cdot\\:1\\cdot\\:\\sec^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=2\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ht1jgkupea8+jUgT+M/VFPgkIqVBCzsTrV5i/ECNCw8tOtZYwUjyXhDTsNnn6Elr8U8lPA4O6WXezAN1J2P7Ue6vfsn/ogOG+fHMfNQh9Je7opJQAzFYPpqn/2bLavXutzgLeG7CLgj8aHpc44iMEx91+u64Nzpp2/441mWGcQ/ET1Ehq4mhUZ9DWVx2az0M" } }, { "type": "step", "result": "=2\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Md1SJKHYLxhOYRRbsLTu+fjbMbEoNNbIR/7Q84OynCfTLx8mOdHYVzxX643JqKFI1Jetp9AvoQcY2QSDwFJhz1kUvGSawlk7umQ8kurx5doE3SiHEhDlxIXJDVY+PPdvgQUxJPyUNnGfVirkcwpVO6RMkE3rByM/Y1Dxxfs3xFw6FFV/0Hm6Y1hI7VaPzzKhkSoZP7CyArJlOo0FRny79Q==" } }, { "type": "step", "primary": "Add similar elements: $$\\sec^{4}\\left(θ\\right)+2\\sec^{4}\\left(θ\\right)=3\\sec^{4}\\left(θ\\right)$$", "result": "=3\\sec^{4}\\left(θ\\right)-2\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ARkhlGZdazrdxNLzhlqePtRiXXTzenxbt00ahL9sDS5L0MjjUe01mdeSwRMoYPSBLTrWWMFI8l4Q07DZ5+hJazaxvejhvTJvqoS105+T9+xc3qH/hVuCay9igw/1IwZBNySLUGdPDJeW3uY1XSs479bA+zX4bD3u3gx65o2NJhP8KqIsxz0lwScBMkZS31XhRCRwG0yFaMyhS3uC5/uUeMvqafR00cq/gMXQBnztqPn0yTzBsNnlyCSr/zrcxdQG" } } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7FkSOkw7Nc+ISjlBZ7XiESO/dTLaj0YrNSw6+/ogyGX8k+jBZ4XxujMMeS3Qvsi6m7aEe0hvT9VtxXei653t2zQi9Vnn4imguIoWn3N1eqjdGIkb3QW2w7y4N83RSktbc/kUupyp7xXxB8YmKEQ5HoPs4SAUa1zTfp2gkDWnxoYVtNtNgAtA4xHxju0w39zFkZEt3ZXAiqUE0HIXrrrezJJi0cdkgnw2HPjRVpcHMxZbIciYVzjaAolEaNkVONtBBRppTNgMFons9bMkTzIHooA==" } } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7aTuTexD/CTl68iqGJang3fkdSragnZ5K8+WMRM5qjJ6ETkUamdUz59t0aHpV3uN0rE0Ib0rKQ2oM97vl/gDbowCWKUbvV6WK3fDUgFtg3Q8XyqIsLZjT7Df8hUObtCZb+5k7zGGz9wmA2lyQsVWy+j6F2h9EjPuS9/S594y1Nd1FKk3fejFkyiOiq9iG9IkAIuSmJUDtLxRPyJ57Nkq7qlZOK6n0rXJ0Nc64aGIzHiSsVGYLehU5hUh50bKLZKnpo1d4a+pBCGznOTLxP2sni3enTnyZLeuZs6pBwuQMplUkt3WiGR7ZaCaXvz77bMjS" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)=12\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(12\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=12\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the Product Rule: $$\\left(f{\\cdot}g\\right)'=f'{\\cdot}g+f{\\cdot}g'$$", "secondary": [ "$$f=\\sec^{4}\\left(θ\\right),\\:g=\\tan\\left(θ\\right)$$" ], "result": "=12\\left(\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)\\tan\\left(θ\\right)+\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)\\sec^{4}\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Product%20Rule", "practiceTopic": "Product Rule" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)=4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{4},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{4}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{4}\\right)=4u^{3}$$", "input": "\\frac{d}{du}\\left(u^{4}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=4u^{4-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=4u^{3}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYl3oEe4EC6n4B8Xgz0cKnkqk3hxk9aCfAWodBRxXgUexuzo0Fl6VhneutEteidn+Kv8//6/nV5O4fb8Xgwi7mapNFxUvwBeni+JEIFAdbegzYEOcQgyW4n51cR7AeMAm7Q==" } }, { "type": "step", "result": "=4u^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgA2CO5cm49xNEqFV0DMaZJRqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidVduWEnoDYDg06IbaTsp9TFtqiuNhFTvmF8ZCDqWZNoubfnOah8gAvLMKOeNMctC5tVYZ/3r82LxPimKUFXjQFStvD8GKDNJJN8qiu0XRcVZAYAvbRKqkRVDaDBHkSBtDwYHkfmHHTze3JXk3LjiJp" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$4\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "4\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{3+1}\\left(θ\\right)$$" ], "result": "=4\\sec^{3+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$3+1=4$$", "result": "=4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7p9w76+/nvHitDrMIupHrrrPmAPxuc/RFHWXkB8A+mnqLeJKmeakO1+Q3aSIldG41PCyA7ixjCdfDYe+yac7KOvOIrkYdKW3Q58EnXaMU+7oslHPSvPf5UfEs5T+IFZ4XHjb2+5NLFZrsH9fcPWg/TXL71sHPAsAQtYZvjpaGSRwboK/Igc1IRC3EGruCoRzV1Zrc/SYRIR8JuxYpD+PJ+Q==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "result": "=\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYs0Vsx3c0nt53Utu5DgHTe3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK00HV/bLwwglXz/2YNx2i4JHggSrmCoGjSdud4CyOvVOgdQN0k+N3NHNAkEIoLrhB631k80nn+ODwI4ID0kCF5+z7zRz81SqRD02HEUE4Wqh" } }, { "type": "step", "result": "=12\\left(4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Simplify $$12\\left(4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right):{\\quad}12\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)$$", "input": "12\\left(4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right)", "result": "=12\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "$$4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)=4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)$$", "input": "4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\tan\\left(θ\\right)\\tan\\left(θ\\right)=\\:\\tan^{1+1}\\left(θ\\right)$$" ], "result": "=4\\sec^{4}\\left(θ\\right)\\tan^{1+1}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7GHerUg4zB8o0EeBO6BKV16xUZgt6FTmFSHnRsotkqemLeJKmeakO1+Q3aSIldG41PCyA7ixjCdfDYe+yac7KOvOIrkYdKW3Q58EnXaMU+7pALhShbMGZIZsZWTO721HsXaBT4KNKuhZZSFsFWnKRStdNJOkRdCtrWB4iMQVmMwBoAWE3NiWlLPrDNnXLOZLK3tKI9zGyi4xCTaccmboyHcRPUSGriaFRn0NZXHZrPQw=" } }, { "type": "interim", "title": "$$\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)=\\sec^{6}\\left(θ\\right)$$", "input": "\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)=\\:\\sec^{2+4}\\left(θ\\right)$$" ], "result": "=\\sec^{2+4}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+4=6$$", "result": "=\\sec^{6}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7AtrsReczPdUqIdn5/Ev5mLBxohAhN47qWD2nlwOHqfkgJ/ZZA32ZInFBpDtxBfiKn3Jyj2VJgBojKQHjINpkkG6JHhxn2McSW7OxLgQBi64O/U6qyRZonn1WPQ/21pz3xMe3sR8ZMoA2Ynlp2W853t3OcQiSW6rZ4zI81UhADI4kt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "=12\\left(\\sec^{6}\\left(θ\\right)+4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7lv52tOJZUIL6vqIybdOpThiMJmjxigW8LortZKvrqEbWPF6oWuXggupsFvXrooIqxMe3sR8ZMoA2Ynlp2W853tMvHyY50dhXPFfrjcmooUinnaTA6rKFuYRlEeyjdJSquO1NenZJP60hEwBH05kmlvOIrkYdKW3Q58EnXaMU+7r4ZAjrJM4mwvgfYovsxoopgQUxJPyUNnGfVirkcwpVO48SyOdzKmRYNaCIczIhnoIh0PvFonSnuZi7Z6ItIhmYhYbzXwn7iok10o2w+NJLRiZIb9CFuBJdPyi+MBoLCMd/LnYygZMYhjzpUCXiHwCy" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=4\\left(-4\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)+12\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(12\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)\\right)=12\\left(64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+132\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+14\\sec^{8}\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(12\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=12\\frac{d}{dθ}\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=12\\left(\\frac{d}{dθ}\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)\\right)+\\frac{d}{dθ}\\left(14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)\\right)=16\\left(4\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+3\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(16\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=16\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\tan^{3}\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the Product Rule: $$\\left(f{\\cdot}g\\right)'=f'{\\cdot}g+f{\\cdot}g'$$", "secondary": [ "$$f=\\sec^{4}\\left(θ\\right),\\:g=\\tan^{3}\\left(θ\\right)$$" ], "result": "=16\\left(\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)\\tan^{3}\\left(θ\\right)+\\frac{d}{dθ}\\left(\\tan^{3}\\left(θ\\right)\\right)\\sec^{4}\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Product%20Rule", "practiceTopic": "Product Rule" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)=4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{4},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{4}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{4}\\right)=4u^{3}$$", "input": "\\frac{d}{du}\\left(u^{4}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=4u^{4-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=4u^{3}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYl3oEe4EC6n4B8Xgz0cKnkqk3hxk9aCfAWodBRxXgUexuzo0Fl6VhneutEteidn+Kv8//6/nV5O4fb8Xgwi7mapNFxUvwBeni+JEIFAdbegzYEOcQgyW4n51cR7AeMAm7Q==" } }, { "type": "step", "result": "=4u^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgA2CO5cm49xNEqFV0DMaZJRqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidVduWEnoDYDg06IbaTsp9TFtqiuNhFTvmF8ZCDqWZNoubfnOah8gAvLMKOeNMctC5tVYZ/3r82LxPimKUFXjQFStvD8GKDNJJN8qiu0XRcVZAYAvbRKqkRVDaDBHkSBtDwYHkfmHHTze3JXk3LjiJp" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$4\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "4\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{3+1}\\left(θ\\right)$$" ], "result": "=4\\sec^{3+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$3+1=4$$", "result": "=4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7p9w76+/nvHitDrMIupHrrrPmAPxuc/RFHWXkB8A+mnqLeJKmeakO1+Q3aSIldG41PCyA7ixjCdfDYe+yac7KOvOIrkYdKW3Q58EnXaMU+7oslHPSvPf5UfEs5T+IFZ4XHjb2+5NLFZrsH9fcPWg/TXL71sHPAsAQtYZvjpaGSRwboK/Igc1IRC3EGruCoRzV1Zrc/SYRIR8JuxYpD+PJ+Q==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan^{3}\\left(θ\\right)\\right)=3\\tan^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan^{3}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}3\\left(\\tan\\left(θ\\right)\\right)^{2}\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan^{3}\\left(θ\\right)\\right)", "result": "=3\\left(\\tan\\left(θ\\right)\\right)^{2}\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{3},\\:\\:u=\\tan\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{3}\\right)\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{3}\\right)=3u^{2}$$", "input": "\\frac{d}{du}\\left(u^{3}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=3u^{3-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=3u^{2}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYjNrXZy15fg+DDm4IZ/kmJKk3hxk9aCfAWodBRxXgUexYrhQEWwJJMG//0JI2CMZRv8//6/nV5O4fb8Xgwi7mapNFxUvwBeni+JEIFAdbegz8nAGF3XMCO2i5cp0uYL6iA==" } }, { "type": "step", "result": "=3u^{2}\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\tan\\left(θ\\right)$$", "result": "=3\\left(\\tan\\left(θ\\right)\\right)^{2}\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYux7TmDJ3fopnHGBxoqO+pYseUrH1bTMvpsf9SfQ9YpVqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKifDI4IHJ+buH5M+0PHLeGB4Be4N1NohQPDRW/BmsNJaPsq1KfCFKwSwA0qmWovUVP9tVYZ/3r82LxPimKUFXjQFStvD8GKDNJJN8qiu0XRcVZAYAvbRKqkRVDaDBHkSBtDwYHkfmHHTze3JXk3LjiJp" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "result": "=\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYs0Vsx3c0nt53Utu5DgHTe3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK00HV/bLwwglXz/2YNx2i4JHggSrmCoGjSdud4CyOvVOgdQN0k+N3NHNAkEIoLrhB631k80nn+ODwI4ID0kCF5+z7zRz81SqRD02HEUE4Wqh" } }, { "type": "step", "result": "=3\\left(\\tan\\left(θ\\right)\\right)^{2}\\sec^{2}\\left(θ\\right)" }, { "type": "step", "primary": "Simplify", "result": "=3\\tan^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=16\\left(4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{3}\\left(θ\\right)+3\\tan^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Simplify $$16\\left(4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{3}\\left(θ\\right)+3\\tan^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right):{\\quad}16\\left(4\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+3\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)$$", "input": "16\\left(4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{3}\\left(θ\\right)+3\\tan^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right)", "result": "=16\\left(4\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+3\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "$$4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{3}\\left(θ\\right)=4\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)$$", "input": "4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{3}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\tan\\left(θ\\right)\\tan^{3}\\left(θ\\right)=\\:\\tan^{1+3}\\left(θ\\right)$$" ], "result": "=4\\sec^{4}\\left(θ\\right)\\tan^{1+3}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+3=4$$", "result": "=4\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7GHerUg4zB8o0EeBO6BKV162cmBc6Js8szDUhC2HxdqjUgkMhq/lXB9qI3pCS5SJMq47vuWedXv2WUg94ER8IwVADhNIEz39BpXewdhMCBb8aab3UY3H06dpMJimNIyD5kuHx4ulZqfxZHIsiQUpV1nyo9elyUFFpcPZW02cAt7EdWYTaqPpaDtAHPW/SgMNicUwY1HWd9EJWOCb4O+kauAND0fAz408eCOOnBAkZ69Akt3WiGR7ZaCaXvz77bMjS" } }, { "type": "interim", "title": "$$3\\tan^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)=3\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)$$", "input": "3\\tan^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)=\\:\\sec^{2+4}\\left(θ\\right)$$" ], "result": "=3\\tan^{2}\\left(θ\\right)\\sec^{2+4}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+4=6$$", "result": "=3\\tan^{2}\\left(θ\\right)\\sec^{6}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7xud7SC9MT1MQdbTGBqrMw+H0qCTik4j9uS7jTF7ng2/8DCHl2/1cUFH2kQZvGLuVzRqDxPUzBN6vjj5oJL9kUEngkySEzEl6diiSTBzPnLnWxcLMxZwB9cGBLGbwbyjTrqr1t4QBx+t+Vf7xBGCGrW9B4jZ4XSgntIkqjxqF2Bvh9Kgk4pOI/bku40xe54Nv6RICywOwLOWusKkGpuhUtYx6/teVLREoXphSPTLZ2MbET1Ehq4mhUZ9DWVx2az0M" } }, { "type": "step", "result": "=16\\left(3\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+4\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7C43rUQmr5KAYI3UGEng6yRiMJmjxigW8LortZKvrqEYERpCXXFEE4hLCZfTeroF6V7JvkhZ2EFhPsvL8iYbkgPpadcGORE5bHX37Io3viWolp/MB1OadflBmLQoq5Pmlo5FYteSPKwXny4uCMrdsKwS9KcmQjztCcdjD5gRjJOz1a03PQ0UrDYBem+/aln9X84iuRh0pbdDnwSddoxT7uh96fmgcT2t23FrnniYZ1CmBBTEk/JQ2cZ9WKuRzClU7UxxUEaCmP0MbiSArdWRj6iHQ+8WidKe5mLtnoi0iGZhpK7ceo/th4B1fP0TRnZr9A1c/UDX6QxYlZyg/KYfMkDSei4QZxxrazp/cx7cYZvYA0hWLB/nt1m4ggrh+mxlbvzIPeEtDfcHv/z8uls8Teg==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)=14\\left(6\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{8}\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(14\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=14\\frac{d}{dθ}\\left(\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the Product Rule: $$\\left(f{\\cdot}g\\right)'=f'{\\cdot}g+f{\\cdot}g'$$", "secondary": [ "$$f=\\sec^{6}\\left(θ\\right),\\:g=\\tan\\left(θ\\right)$$" ], "result": "=14\\left(\\frac{d}{dθ}\\left(\\sec^{6}\\left(θ\\right)\\right)\\tan\\left(θ\\right)+\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)\\sec^{6}\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Product%20Rule", "practiceTopic": "Product Rule" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec^{6}\\left(θ\\right)\\right)=6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{6}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}6\\left(\\sec\\left(θ\\right)\\right)^{5}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{6}\\left(θ\\right)\\right)", "result": "=6\\left(\\sec\\left(θ\\right)\\right)^{5}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{6},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{6}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{6}\\right)=6u^{5}$$", "input": "\\frac{d}{du}\\left(u^{6}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=6u^{6-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=6u^{5}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYgOK76JOSNqGa1B1wU6ozWWk3hxk9aCfAWodBRxXgUexqSoDkuZ4Ab1St3bM8oqKcP8//6/nV5O4fb8Xgwi7mapNFxUvwBeni+JEIFAdbegzHfwjUTnSu4cWqw+8m79+NA==" } }, { "type": "step", "result": "=6u^{5}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=6\\left(\\sec\\left(θ\\right)\\right)^{5}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgBaJTGF5uPwSMzxoSyJNVZsqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKieMSe35zYqdvyntnEYxaQ88jJq7JZFtKgvxzzzuaqTGrebfnOah8gAvLMKOeNMctC5tVYZ/3r82LxPimKUFXjQFStvD8GKDNJJN8qiu0XRcVZAYAvbRKqkRVDaDBHkSBtDwYHkfmHHTze3JXk3LjiJp" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=6\\left(\\sec\\left(θ\\right)\\right)^{5}\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$6\\sec^{5}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "6\\sec^{5}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{5}\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{5+1}\\left(θ\\right)$$" ], "result": "=6\\sec^{5+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$5+1=6$$", "result": "=6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7QMJmgA2Sx2hYpGS5j/h1NLPmAPxuc/RFHWXkB8A+mnqLeJKmeakO1+Q3aSIldG417JiujyaxykLR7Tv0Nr+wOGprXgPFMmP3i5vY2oD+e4YslHPSvPf5UfEs5T+IFZ4XHjb2+5NLFZrsH9fcPWg/TRiIlnZT/wCWlhB0uSVNtvx/uR/EZqc6EbI3tnxSI5001Zrc/SYRIR8JuxYpD+PJ+Q==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "result": "=\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYs0Vsx3c0nt53Utu5DgHTe3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK00HV/bLwwglXz/2YNx2i4JHggSrmCoGjSdud4CyOvVOgdQN0k+N3NHNAkEIoLrhB631k80nn+ODwI4ID0kCF5+z7zRz81SqRD02HEUE4Wqh" } }, { "type": "step", "result": "=14\\left(6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{6}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Simplify $$14\\left(6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{6}\\left(θ\\right)\\right):{\\quad}14\\left(6\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{8}\\left(θ\\right)\\right)$$", "input": "14\\left(6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{6}\\left(θ\\right)\\right)", "result": "=14\\left(6\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{8}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "$$6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)=6\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)$$", "input": "6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\tan\\left(θ\\right)\\tan\\left(θ\\right)=\\:\\tan^{1+1}\\left(θ\\right)$$" ], "result": "=6\\sec^{6}\\left(θ\\right)\\tan^{1+1}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=6\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7H4jwT8QhunkiDAGNb+RcvaxUZgt6FTmFSHnRsotkqemLeJKmeakO1+Q3aSIldG417JiujyaxykLR7Tv0Nr+wOGprXgPFMmP3i5vY2oD+e4ZALhShbMGZIZsZWTO721Hsn1G20l7DmbyzInDIeLRU/ZYojigZ8i1ymB3KwWBswnjgWFmp7EeEDyNH2q8Tm1k1jHr+15UtEShemFI9MtnYxsRPUSGriaFRn0NZXHZrPQw=" } }, { "type": "interim", "title": "$$\\sec^{2}\\left(θ\\right)\\sec^{6}\\left(θ\\right)=\\sec^{8}\\left(θ\\right)$$", "input": "\\sec^{2}\\left(θ\\right)\\sec^{6}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{2}\\left(θ\\right)\\sec^{6}\\left(θ\\right)=\\:\\sec^{2+6}\\left(θ\\right)$$" ], "result": "=\\sec^{2+6}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+6=8$$", "result": "=\\sec^{8}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7AtrsReczPdUqIdn5/Ev5mH+O3FtTTH45VMxFH8DJ4BEgJ/ZZA32ZInFBpDtxBfiKn3Jyj2VJgBojKQHjINpkkI6d/oVrekw7T/Cky7x1KxcO/U6qyRZonn1WPQ/21pz3HK+miBBOGkRkZzIN3nn5xFhBmsSvLw1yh3wbSiSoUDgkt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "=14\\left(\\sec^{8}\\left(θ\\right)+6\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7nO0mFYRqkI6v7Sy9VTWwjRiMJmjxigW8LortZKvrqEbWPF6oWuXggupsFvXrooIqHK+miBBOGkRkZzIN3nn5xNMvHyY50dhXPFfrjcmooUiPqGSxzi9gRanHsLmpQgKwsfoApmSO56Ak/1ybzTsFLmprXgPFMmP3i5vY2oD+e4b4ZAjrJM4mwvgfYovsxoopgQUxJPyUNnGfVirkcwpVO2TUlQSCHNJw/4wJ8m9ormYqoLNuWbSjE1JT+hrPzX/RhYbzXwn7iok10o2w+NJLRiZIb9CFuBJdPyi+MBoLCMeBkcS/lgaNKtB6zcW1E3BB" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=12\\left(16\\left(4\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+3\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)+14\\left(6\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{8}\\left(θ\\right)\\right)\\right)" }, { "type": "interim", "title": "Expand $$16\\left(4\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+3\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)+14\\left(6\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{8}\\left(θ\\right)\\right):{\\quad}64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+132\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+14\\sec^{8}\\left(θ\\right)$$", "input": "16\\left(4\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+3\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)+14\\left(6\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{8}\\left(θ\\right)\\right)", "result": "=12\\left(64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+132\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+14\\sec^{8}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Expand $$16\\left(4\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+3\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right):{\\quad}64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+48\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)$$", "input": "16\\left(4\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+3\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)", "result": "=64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+48\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+14\\left(6\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{8}\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b+c\\right)=ab+ac$$", "secondary": [ "$$a=16,\\:b=4\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right),\\:c=3\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)$$" ], "result": "=16\\cdot\\:4\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+16\\cdot\\:3\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "interim", "title": "Simplify $$16\\cdot\\:4\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+16\\cdot\\:3\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right):{\\quad}64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+48\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)$$", "input": "16\\cdot\\:4\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+16\\cdot\\:3\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)", "result": "=64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+48\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$16\\cdot\\:4=64$$", "result": "=64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+16\\cdot\\:3\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)" }, { "type": "step", "primary": "Multiply the numbers: $$16\\cdot\\:3=48$$", "result": "=64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+48\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7C43rUQmr5KAYI3UGEng6yV6kSs12eASfWmxynRkYZ3imWJLfEdyBmZh9L6HIlMnFSPKgibEOx6xAH14rsVQdDd6GQqufR6tr2vPxOUv7H+/wV+6wolBQOY9LPugwYJ6C3tKI9zGyi4xCTaccmboyHVWGy3JBiLa3A/i0ciCokDrRsdR53zGMNMouUmBvz53xupxJf7boNv1S9Hc0DXJXvt7UD16RqFQha281as3i/3gIhVb1SjznyQ5krYRiFqNmoU0M/hfYOSS/FiVCoXaNrXxGVgveSDjBoBB0CLM8z96ArlFiGY8qAdw44hDradHvvzIPeEtDfcHv/z8uls8Teg==" } }, { "type": "interim", "title": "Expand $$14\\left(6\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{8}\\left(θ\\right)\\right):{\\quad}84\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+14\\sec^{8}\\left(θ\\right)$$", "input": "14\\left(6\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{8}\\left(θ\\right)\\right)", "result": "=64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+48\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+84\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+14\\sec^{8}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b+c\\right)=ab+ac$$", "secondary": [ "$$a=14,\\:b=6\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right),\\:c=\\sec^{8}\\left(θ\\right)$$" ], "result": "=14\\cdot\\:6\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+14\\sec^{8}\\left(θ\\right)", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "primary": "Multiply the numbers: $$14\\cdot\\:6=84$$", "result": "=84\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+14\\sec^{8}\\left(θ\\right)" } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7nO0mFYRqkI6v7Sy9VTWwjSvxearhctbAPxX3QGBh5kkx0tWj1vW7tIvP14Mwoy6mLTrWWMFI8l4Q07DZ5+hJazPkBMIuMrWSeavgWBoF37pgEq4U4c+VEIUu/cb6SEfLgBHHd6NlsvdNQCpoDejgKWxRFK5IwscCsNiX9YX0G6d6pfF1z6umzUJTJvt+ojYZ00PxkCV/xFqV0mmm+Nwrth1Y04xtNK2BCM2tE3PxmDme5tn70JIT2jKyMskERPvitGNYJGuT8kFILfu1szjq+A==" } }, { "type": "step", "primary": "Add similar elements: $$48\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+84\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)=132\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)$$", "result": "=64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+132\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+14\\sec^{8}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7C43rUQmr5KAYI3UGEng6yV6kSs12eASfWmxynRkYZ3imWJLfEdyBmZh9L6HIlMnFquwP5S7dVU2hOl0jx4UuGyqgs25ZtKMTUlP6Gs/Nf9EOOEJ8eYI1Z69ODewOg6d0GN3NaV3za5y0fd8sDMMqbt13jtrSFDx+UNsawjlOjV0T07YNGlOvsbUamOsS8Bedjha9s6alqYoo6ur9CV5YJR67hL3SLnd3F8UQKch9b3y7ssmRiNW3dHLSNd768UtCbqAWk+sYoFfdaspteys3OK6q9beEAcfrflX+8QRghq2CcvL2jSzw1VlRcGX4ka6qUYDxHhHJhPwBywaMxzqJZlGGuUmWLwacJdjIdVxl7S4IFLNGVGiL7NHRmdqGkv+kzFx+puj8P4X++28hfOTEK4y85Xs96ub4hnNPpGMCnfiCWjNEoRZwVGhuwKG+ZGEU32vN1Q0V3hnPq2BOOEcCJA==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=8\\left(-4\\left(-4\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)+12\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)\\right)+12\\left(64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+132\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+14\\sec^{8}\\left(θ\\right)\\right)\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(24\\left(256\\tan^{5}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+1048\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+376\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)\\right)\\right)=24\\left(7568\\sec^{6}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+1024\\tan^{6}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+6152\\sec^{8}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+376\\sec^{10}\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(24\\left(256\\tan^{5}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+1048\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+376\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=24\\frac{d}{dθ}\\left(256\\tan^{5}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+1048\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)+376\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=24\\left(\\frac{d}{dθ}\\left(256\\tan^{5}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right)+\\frac{d}{dθ}\\left(1048\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)\\right)+\\frac{d}{dθ}\\left(376\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(256\\tan^{5}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right)=256\\left(5\\sec^{6}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+4\\tan^{6}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(256\\tan^{5}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=256\\frac{d}{dθ}\\left(\\tan^{5}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the Product Rule: $$\\left(f{\\cdot}g\\right)'=f'{\\cdot}g+f{\\cdot}g'$$", "secondary": [ "$$f=\\tan^{5}\\left(θ\\right),\\:g=\\sec^{4}\\left(θ\\right)$$" ], "result": "=256\\left(\\frac{d}{dθ}\\left(\\tan^{5}\\left(θ\\right)\\right)\\sec^{4}\\left(θ\\right)+\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)\\tan^{5}\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Product%20Rule", "practiceTopic": "Product Rule" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan^{5}\\left(θ\\right)\\right)=5\\tan^{4}\\left(θ\\right)\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan^{5}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}5\\left(\\tan\\left(θ\\right)\\right)^{4}\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan^{5}\\left(θ\\right)\\right)", "result": "=5\\left(\\tan\\left(θ\\right)\\right)^{4}\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{5},\\:\\:u=\\tan\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{5}\\right)\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{5}\\right)=5u^{4}$$", "input": "\\frac{d}{du}\\left(u^{5}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=5u^{5-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=5u^{4}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYlQRLITLpaEyfaA4k5qpTXCk3hxk9aCfAWodBRxXgUexxoTeBEwHv56wOfn5XxqhLP8//6/nV5O4fb8Xgwi7mapNFxUvwBeni+JEIFAdbegzDDdkKACXWyFAKMUXsRKp6w==" } }, { "type": "step", "result": "=5u^{4}\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\tan\\left(θ\\right)$$", "result": "=5\\left(\\tan\\left(θ\\right)\\right)^{4}\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYux7TmDJ3fopnHGBxoqO+pZtVgwqk46dBr/619dmMqkaqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKifQfjs9pZVnewix0bdSyDWsbU5YsQtLgFNTWf9EGqSvuMq1KfCFKwSwA0qmWovUVP9tVYZ/3r82LxPimKUFXjQFStvD8GKDNJJN8qiu0XRcVZAYAvbRKqkRVDaDBHkSBtDwYHkfmHHTze3JXk3LjiJp" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "result": "=\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYs0Vsx3c0nt53Utu5DgHTe3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK00HV/bLwwglXz/2YNx2i4JHggSrmCoGjSdud4CyOvVOgdQN0k+N3NHNAkEIoLrhB631k80nn+ODwI4ID0kCF5+z7zRz81SqRD02HEUE4Wqh" } }, { "type": "step", "result": "=5\\left(\\tan\\left(θ\\right)\\right)^{4}\\sec^{2}\\left(θ\\right)" }, { "type": "step", "primary": "Simplify", "result": "=5\\tan^{4}\\left(θ\\right)\\sec^{2}\\left(θ\\right)", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)=4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{4}\\left(θ\\right)\\right)", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{4},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{4}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{4}\\right)=4u^{3}$$", "input": "\\frac{d}{du}\\left(u^{4}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=4u^{4-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=4u^{3}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYl3oEe4EC6n4B8Xgz0cKnkqk3hxk9aCfAWodBRxXgUexuzo0Fl6VhneutEteidn+Kv8//6/nV5O4fb8Xgwi7mapNFxUvwBeni+JEIFAdbegzYEOcQgyW4n51cR7AeMAm7Q==" } }, { "type": "step", "result": "=4u^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgA2CO5cm49xNEqFV0DMaZJRqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidVduWEnoDYDg06IbaTsp9TFtqiuNhFTvmF8ZCDqWZNoubfnOah8gAvLMKOeNMctC5tVYZ/3r82LxPimKUFXjQFStvD8GKDNJJN8qiu0XRcVZAYAvbRKqkRVDaDBHkSBtDwYHkfmHHTze3JXk3LjiJp" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=4\\left(\\sec\\left(θ\\right)\\right)^{3}\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$4\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "4\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{3}\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{3+1}\\left(θ\\right)$$" ], "result": "=4\\sec^{3+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$3+1=4$$", "result": "=4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7p9w76+/nvHitDrMIupHrrrPmAPxuc/RFHWXkB8A+mnqLeJKmeakO1+Q3aSIldG41PCyA7ixjCdfDYe+yac7KOvOIrkYdKW3Q58EnXaMU+7oslHPSvPf5UfEs5T+IFZ4XHjb2+5NLFZrsH9fcPWg/TXL71sHPAsAQtYZvjpaGSRwboK/Igc1IRC3EGruCoRzV1Zrc/SYRIR8JuxYpD+PJ+Q==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=256\\left(5\\tan^{4}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{5}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Simplify $$256\\left(5\\tan^{4}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{5}\\left(θ\\right)\\right):{\\quad}256\\left(5\\sec^{6}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+4\\tan^{6}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right)$$", "input": "256\\left(5\\tan^{4}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{5}\\left(θ\\right)\\right)", "result": "=256\\left(5\\sec^{6}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+4\\tan^{6}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "$$5\\tan^{4}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)=5\\sec^{6}\\left(θ\\right)\\tan^{4}\\left(θ\\right)$$", "input": "5\\tan^{4}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{2}\\left(θ\\right)\\sec^{4}\\left(θ\\right)=\\:\\sec^{2+4}\\left(θ\\right)$$" ], "result": "=5\\tan^{4}\\left(θ\\right)\\sec^{2+4}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+4=6$$", "result": "=5\\tan^{4}\\left(θ\\right)\\sec^{6}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7kNaRv8Wne71UtPwtJRfQE+H0qCTik4j9uS7jTF7ng2/8DCHl2/1cUFH2kQZvGLuVzRqDxPUzBN6vjj5oJL9kUFtyWBEiTWETiygsuqEJ16jWxcLMxZwB9cGBLGbwbyjTrqr1t4QBx+t+Vf7xBGCGrcahlpq6JPpgIQKz4KWo36nh9Kgk4pOI/bku40xe54Nv2WHdFm+5lfN+r0Gs7w1l9Ix6/teVLREoXphSPTLZ2MY+TTtd7+vBzI9nKngLCVse" } }, { "type": "interim", "title": "$$4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{5}\\left(θ\\right)=4\\tan^{6}\\left(θ\\right)\\sec^{4}\\left(θ\\right)$$", "input": "4\\sec^{4}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{5}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\tan\\left(θ\\right)\\tan^{5}\\left(θ\\right)=\\:\\tan^{1+5}\\left(θ\\right)$$" ], "result": "=4\\sec^{4}\\left(θ\\right)\\tan^{1+5}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+5=6$$", "result": "=4\\sec^{4}\\left(θ\\right)\\tan^{6}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7GHerUg4zB8o0EeBO6BKV162cmBc6Js8szDUhC2HxdqhNN0U/i6wQcI1lw3i9ZSm1q47vuWedXv2WUg94ER8IwVADhNIEz39BpXewdhMCBb9R6ajCC1RKdE2e/XsfgLSckuHx4ulZqfxZHIsiQUpV1nyo9elyUFFpcPZW02cAt7HAjSJgiGZ+diyKG5Zhi1XpFun9srfULSXJeGLSNXDEhGAEODKzAB5RvbiyYeFhfmIkt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "=256\\left(5\\sec^{6}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+4\\tan^{6}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s70fRHDyniztXQBgfpnrvazjSei4QZxxrazp/cx7cYZvbsZrmODjX3kWJgP0e3FlGT84iuRh0pbdDnwSddoxT7umppni78657t6cHZMpr1bnF+EoxXcOMsKEq/MwpvwRA+zMFYmi1F5Hg/ibpEToVnYwTxkKDH0HybCSYeTZoQsetepErNdngEn1pscp0ZGGd4zYM2aL8m2UMI4UKpbjUA5mi+1rvdbLZkXWczz9SdWfVFKk3fejFkyiOiq9iG9IkAIuSmJUDtLxRPyJ57Nkq7qvO8ToBBG+V4IJAWgjU+uWZzxRz/FgXclHQ8HmHckcWwbB3+GenhKNoSiFNUUnFm4+jVKmAf6o6tJxDt2GB4spuozCu/C2UyLX0IYpGschsdtmRi0E3eC0He/xESpULhVg==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(1048\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)\\right)=1048\\left(6\\sec^{6}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+3\\sec^{8}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(1048\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=1048\\frac{d}{dθ}\\left(\\sec^{6}\\left(θ\\right)\\tan^{3}\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the Product Rule: $$\\left(f{\\cdot}g\\right)'=f'{\\cdot}g+f{\\cdot}g'$$", "secondary": [ "$$f=\\sec^{6}\\left(θ\\right),\\:g=\\tan^{3}\\left(θ\\right)$$" ], "result": "=1048\\left(\\frac{d}{dθ}\\left(\\sec^{6}\\left(θ\\right)\\right)\\tan^{3}\\left(θ\\right)+\\frac{d}{dθ}\\left(\\tan^{3}\\left(θ\\right)\\right)\\sec^{6}\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Product%20Rule", "practiceTopic": "Product Rule" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec^{6}\\left(θ\\right)\\right)=6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{6}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}6\\left(\\sec\\left(θ\\right)\\right)^{5}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{6}\\left(θ\\right)\\right)", "result": "=6\\left(\\sec\\left(θ\\right)\\right)^{5}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{6},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{6}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{6}\\right)=6u^{5}$$", "input": "\\frac{d}{du}\\left(u^{6}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=6u^{6-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=6u^{5}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYgOK76JOSNqGa1B1wU6ozWWk3hxk9aCfAWodBRxXgUexqSoDkuZ4Ab1St3bM8oqKcP8//6/nV5O4fb8Xgwi7mapNFxUvwBeni+JEIFAdbegzHfwjUTnSu4cWqw+8m79+NA==" } }, { "type": "step", "result": "=6u^{5}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=6\\left(\\sec\\left(θ\\right)\\right)^{5}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgBaJTGF5uPwSMzxoSyJNVZsqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKieMSe35zYqdvyntnEYxaQ88jJq7JZFtKgvxzzzuaqTGrebfnOah8gAvLMKOeNMctC5tVYZ/3r82LxPimKUFXjQFStvD8GKDNJJN8qiu0XRcVZAYAvbRKqkRVDaDBHkSBtDwYHkfmHHTze3JXk3LjiJp" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=6\\left(\\sec\\left(θ\\right)\\right)^{5}\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$6\\sec^{5}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "6\\sec^{5}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{5}\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{5+1}\\left(θ\\right)$$" ], "result": "=6\\sec^{5+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$5+1=6$$", "result": "=6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7QMJmgA2Sx2hYpGS5j/h1NLPmAPxuc/RFHWXkB8A+mnqLeJKmeakO1+Q3aSIldG417JiujyaxykLR7Tv0Nr+wOGprXgPFMmP3i5vY2oD+e4YslHPSvPf5UfEs5T+IFZ4XHjb2+5NLFZrsH9fcPWg/TRiIlnZT/wCWlhB0uSVNtvx/uR/EZqc6EbI3tnxSI5001Zrc/SYRIR8JuxYpD+PJ+Q==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan^{3}\\left(θ\\right)\\right)=3\\tan^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan^{3}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}3\\left(\\tan\\left(θ\\right)\\right)^{2}\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan^{3}\\left(θ\\right)\\right)", "result": "=3\\left(\\tan\\left(θ\\right)\\right)^{2}\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{3},\\:\\:u=\\tan\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{3}\\right)\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{3}\\right)=3u^{2}$$", "input": "\\frac{d}{du}\\left(u^{3}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=3u^{3-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=3u^{2}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYjNrXZy15fg+DDm4IZ/kmJKk3hxk9aCfAWodBRxXgUexYrhQEWwJJMG//0JI2CMZRv8//6/nV5O4fb8Xgwi7mapNFxUvwBeni+JEIFAdbegz8nAGF3XMCO2i5cp0uYL6iA==" } }, { "type": "step", "result": "=3u^{2}\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\tan\\left(θ\\right)$$", "result": "=3\\left(\\tan\\left(θ\\right)\\right)^{2}\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYux7TmDJ3fopnHGBxoqO+pYseUrH1bTMvpsf9SfQ9YpVqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKifDI4IHJ+buH5M+0PHLeGB4Be4N1NohQPDRW/BmsNJaPsq1KfCFKwSwA0qmWovUVP9tVYZ/3r82LxPimKUFXjQFStvD8GKDNJJN8qiu0XRcVZAYAvbRKqkRVDaDBHkSBtDwYHkfmHHTze3JXk3LjiJp" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "result": "=\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYs0Vsx3c0nt53Utu5DgHTe3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK00HV/bLwwglXz/2YNx2i4JHggSrmCoGjSdud4CyOvVOgdQN0k+N3NHNAkEIoLrhB631k80nn+ODwI4ID0kCF5+z7zRz81SqRD02HEUE4Wqh" } }, { "type": "step", "result": "=3\\left(\\tan\\left(θ\\right)\\right)^{2}\\sec^{2}\\left(θ\\right)" }, { "type": "step", "primary": "Simplify", "result": "=3\\tan^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=1048\\left(6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{3}\\left(θ\\right)+3\\tan^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{6}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Simplify $$1048\\left(6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{3}\\left(θ\\right)+3\\tan^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{6}\\left(θ\\right)\\right):{\\quad}1048\\left(6\\sec^{6}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+3\\sec^{8}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)$$", "input": "1048\\left(6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{3}\\left(θ\\right)+3\\tan^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{6}\\left(θ\\right)\\right)", "result": "=1048\\left(6\\sec^{6}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+3\\sec^{8}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "$$6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{3}\\left(θ\\right)=6\\sec^{6}\\left(θ\\right)\\tan^{4}\\left(θ\\right)$$", "input": "6\\sec^{6}\\left(θ\\right)\\tan\\left(θ\\right)\\tan^{3}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\tan\\left(θ\\right)\\tan^{3}\\left(θ\\right)=\\:\\tan^{1+3}\\left(θ\\right)$$" ], "result": "=6\\sec^{6}\\left(θ\\right)\\tan^{1+3}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+3=4$$", "result": "=6\\sec^{6}\\left(θ\\right)\\tan^{4}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7H4jwT8QhunkiDAGNb+Rcva2cmBc6Js8szDUhC2HxdqjUgkMhq/lXB9qI3pCS5SJMq47vuWedXv2WUg94ER8IwZGREjArFZVzoA4Cy52193kaab3UY3H06dpMJimNIyD5wqP97mnHf8zpjilA0FhQHuj4b/xSLwXZQVzICbe8+7UdWYTaqPpaDtAHPW/SgMNiXXbfFRrmBUC90vtWO60GkgND0fAz408eCOOnBAkZ69Akt3WiGR7ZaCaXvz77bMjS" } }, { "type": "interim", "title": "$$3\\tan^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{6}\\left(θ\\right)=3\\sec^{8}\\left(θ\\right)\\tan^{2}\\left(θ\\right)$$", "input": "3\\tan^{2}\\left(θ\\right)\\sec^{2}\\left(θ\\right)\\sec^{6}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{2}\\left(θ\\right)\\sec^{6}\\left(θ\\right)=\\:\\sec^{2+6}\\left(θ\\right)$$" ], "result": "=3\\tan^{2}\\left(θ\\right)\\sec^{2+6}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+6=8$$", "result": "=3\\tan^{2}\\left(θ\\right)\\sec^{8}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7xud7SC9MT1MQdbTGBqrMw+H0qCTik4j9uS7jTF7ng2+1wTB/eqG+qkXWWKkar/IKzRqDxPUzBN6vjj5oJL9kUEngkySEzEl6diiSTBzPnLlchq3Om7OzPBTE1oy4zepFrqr1t4QBx+t+Vf7xBGCGrW9B4jZ4XSgntIkqjxqF2Bvh9Kgk4pOI/bku40xe54NvUO+MOu9VhHmTE6rfCn6vlY0lC2e/2hYJJh2lKCrBKAvET1Ehq4mhUZ9DWVx2az0M" } }, { "type": "step", "result": "=1048\\left(3\\sec^{8}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+6\\sec^{6}\\left(θ\\right)\\tan^{4}\\left(θ\\right)\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7OJe6Wsm1BFNjIHYVu5/v5wdelOk4fDgtZq7E8gsiQKDyMVfapT21Z107znrKVfuZovAY+Qs1xqG4hI5I9243R+1EoURFS5yzrLcy0QSsgOutf06qhKmWxn5oabF03pUPcJChiVhDxT5N/LHSTLMjyFcTCGWXWg8FmqjcfMdc9TG7ssmRiNW3dHLSNd768UtCy20THSWqZhRacDDCHM8dRICRiN3L3pWFPsR7L9QzkpxkS3dlcCKpQTQcheuut7MkdkqXaZXgCXO2IClMAWKDPcOTPvWED+ayzSyV/npWXRsHXpTpOHw4LWauxPILIkCg8jFX2qU9tWddO856ylX7maLwGPkLNcahuISOSPduN0ftRKFERUucs6y3MtEErIDrSIUZGk+g6gqVwxD/XktSUw==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(376\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)\\right)=376\\left(8\\sec^{8}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{10}\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(376\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=376\\frac{d}{dθ}\\left(\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Apply the Product Rule: $$\\left(f{\\cdot}g\\right)'=f'{\\cdot}g+f{\\cdot}g'$$", "secondary": [ "$$f=\\sec^{8}\\left(θ\\right),\\:g=\\tan\\left(θ\\right)$$" ], "result": "=376\\left(\\frac{d}{dθ}\\left(\\sec^{8}\\left(θ\\right)\\right)\\tan\\left(θ\\right)+\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)\\sec^{8}\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Product%20Rule", "practiceTopic": "Product Rule" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec^{8}\\left(θ\\right)\\right)=8\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{8}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}8\\left(\\sec\\left(θ\\right)\\right)^{7}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec^{8}\\left(θ\\right)\\right)", "result": "=8\\left(\\sec\\left(θ\\right)\\right)^{7}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{8},\\:\\:u=\\sec\\left(θ\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{8}\\right)\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{8}\\right)=8u^{7}$$", "input": "\\frac{d}{du}\\left(u^{8}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=8u^{8-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=8u^{7}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqfCu3R3nHIhDvsPcLLz/1Kk3hxk9aCfAWodBRxXgUexxBsJp2NepDuy/SDgSGtTQP8//6/nV5O4fb8Xgwi7mapNFxUvwBeni+JEIFAdbegz9Cf5/xFJQkGUmQmSfOtWkA==" } }, { "type": "step", "result": "=8u^{7}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(θ\\right)$$", "result": "=8\\left(\\sec\\left(θ\\right)\\right)^{7}\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqomC760uTfi5styIzLqfgAmdV6hJ9cldZkSASENQvDKqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKie/FI4A8N6pVJJZl5fBw9jL2Bb9Z084y+BADz+fW/X+8ObfnOah8gAvLMKOeNMctC5tVYZ/3r82LxPimKUFXjQFStvD8GKDNJJN8qiu0XRcVZAYAvbRKqkRVDaDBHkSBtDwYHkfmHHTze3JXk3LjiJp" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\sec\\left(θ\\right)\\right)=\\sec\\left(θ\\right)\\tan\\left(θ\\right)$$", "result": "=\\sec\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnzvPkSExKIXJeo0bEP5QH3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK7PmAPxuc/RFHWXkB8A+mnquqvW3hAHH635V/vEEYIatejKKUMuY+AoWpcXCVIrjHiszt3YeaDjRHtGWgLaorsaZ8pcHxijGBaQdTcEoewm1JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=8\\left(\\sec\\left(θ\\right)\\right)^{7}\\sec\\left(θ\\right)\\tan\\left(θ\\right)" }, { "type": "interim", "title": "Simplify $$8\\sec^{7}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right):{\\quad}8\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)$$", "input": "8\\sec^{7}\\left(θ\\right)\\sec\\left(θ\\right)\\tan\\left(θ\\right)", "result": "=8\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{7}\\left(θ\\right)\\sec\\left(θ\\right)=\\:\\sec^{7+1}\\left(θ\\right)$$" ], "result": "=8\\sec^{7+1}\\left(θ\\right)\\tan\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$7+1=8$$", "result": "=8\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s788m1NnShpYfas9AbMiew4LPmAPxuc/RFHWXkB8A+mnqLeJKmeakO1+Q3aSIldG41ADbve3/7gGcExW06cBXhP0h3SRMiT3saoUBG0dunv7MslHPSvPf5UfEs5T+IFZ4XHjb2+5NLFZrsH9fcPWg/TWEAQUsScCNiVF2zTM1OQ/b8F+KWgQQNkMfEM+3mCuVT1Zrc/SYRIR8JuxYpD+PJ+Q==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "input": "\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dθ}\\left(\\tan\\left(θ\\right)\\right)=\\sec^{2}\\left(θ\\right)$$", "result": "=\\sec^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYs0Vsx3c0nt53Utu5DgHTe3VOTwum64s2MWy7bE0OnBHo5FYteSPKwXny4uCMrdsK00HV/bLwwglXz/2YNx2i4JHggSrmCoGjSdud4CyOvVOgdQN0k+N3NHNAkEIoLrhB631k80nn+ODwI4ID0kCF5+z7zRz81SqRD02HEUE4Wqh" } }, { "type": "step", "result": "=376\\left(8\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{8}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Simplify $$376\\left(8\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{8}\\left(θ\\right)\\right):{\\quad}376\\left(8\\sec^{8}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{10}\\left(θ\\right)\\right)$$", "input": "376\\left(8\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)+\\sec^{2}\\left(θ\\right)\\sec^{8}\\left(θ\\right)\\right)", "result": "=376\\left(8\\sec^{8}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{10}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "$$8\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)=8\\sec^{8}\\left(θ\\right)\\tan^{2}\\left(θ\\right)$$", "input": "8\\sec^{8}\\left(θ\\right)\\tan\\left(θ\\right)\\tan\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\tan\\left(θ\\right)\\tan\\left(θ\\right)=\\:\\tan^{1+1}\\left(θ\\right)$$" ], "result": "=8\\sec^{8}\\left(θ\\right)\\tan^{1+1}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=8\\sec^{8}\\left(θ\\right)\\tan^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7DQbpSNGbJv3OaxVtSCmYTaxUZgt6FTmFSHnRsotkqemLeJKmeakO1+Q3aSIldG41ADbve3/7gGcExW06cBXhP0h3SRMiT3saoUBG0dunv7NALhShbMGZIZsZWTO721Hs6dA8Zjy0u7hmCe7XQaWlOENJNa40YCP7AjAwax3ssJbXw77XGApYnF1p5sYP331xjSULZ7/aFgkmHaUoKsEoC8RPUSGriaFRn0NZXHZrPQw=" } }, { "type": "interim", "title": "$$\\sec^{2}\\left(θ\\right)\\sec^{8}\\left(θ\\right)=\\sec^{10}\\left(θ\\right)$$", "input": "\\sec^{2}\\left(θ\\right)\\sec^{8}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sec^{2}\\left(θ\\right)\\sec^{8}\\left(θ\\right)=\\:\\sec^{2+8}\\left(θ\\right)$$" ], "result": "=\\sec^{2+8}\\left(θ\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+8=10$$", "result": "=\\sec^{10}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7AtrsReczPdUqIdn5/Ev5mAADl82NcivZS+ngood9TCAgJ/ZZA32ZInFBpDtxBfiKn3Jyj2VJgBojKQHjINpkkIW9LZLhxHePyFtknozK+1iZucCorOwYXmPiiRdW1pk1oNp/T7Lj+fhfJjDBxRSsBn8z64z7zUzynTk2VXXbvZuwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=376\\left(\\sec^{10}\\left(θ\\right)+8\\sec^{8}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7XNVaZKNnv2s7DNmNSSmE4yKI9Km/ZvzCnLF9eqmklKH+eLi8LuyQmbqRSbuaHIeLoNp/T7Lj+fhfJjDBxRSsBvbON/2huEaiIUfMJ+XMukyB5cQUIPnysMUN/Jtl/XoknLn06LH3yHHk856zmPe1fzO+HM0vjNm8E/4Tn1PHA+3Yd4NT1kHhOapXwjflRI8Go3oe/oyhMy2+1TQhDBd2f2zM6E3fuZxF1XkKAYaRXCDChyS+BUO5pBvmNSdOh/HsGIwmaPGKBbwuiu1kq+uoRtY8Xqha5eCC6mwW9euigirr2sxa1dbmsR6c3xVXRBkU4gBJl4WMO1rA0a30/bUYlg==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=24\\left(256\\left(5\\sec^{6}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+4\\tan^{6}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right)+1048\\left(6\\sec^{6}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+3\\sec^{8}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)+376\\left(8\\sec^{8}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{10}\\left(θ\\right)\\right)\\right)" }, { "type": "interim", "title": "Expand $$256\\left(5\\sec^{6}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+4\\tan^{6}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right)+1048\\left(6\\sec^{6}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+3\\sec^{8}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)+376\\left(8\\sec^{8}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{10}\\left(θ\\right)\\right):{\\quad}7568\\sec^{6}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+1024\\tan^{6}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+6152\\sec^{8}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+376\\sec^{10}\\left(θ\\right)$$", "input": "256\\left(5\\sec^{6}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+4\\tan^{6}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right)+1048\\left(6\\sec^{6}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+3\\sec^{8}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)+376\\left(8\\sec^{8}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{10}\\left(θ\\right)\\right)", "result": "=24\\left(7568\\sec^{6}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+1024\\tan^{6}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+6152\\sec^{8}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+376\\sec^{10}\\left(θ\\right)\\right)", "steps": [ { "type": "interim", "title": "Expand $$256\\left(5\\sec^{6}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+4\\tan^{6}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right):{\\quad}1280\\sec^{6}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+1024\\tan^{6}\\left(θ\\right)\\sec^{4}\\left(θ\\right)$$", "input": "256\\left(5\\sec^{6}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+4\\tan^{6}\\left(θ\\right)\\sec^{4}\\left(θ\\right)\\right)", "result": "=1280\\sec^{6}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+1024\\tan^{6}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+1048\\left(6\\sec^{6}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+3\\sec^{8}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)+376\\left(8\\sec^{8}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{10}\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b+c\\right)=ab+ac$$", "secondary": [ "$$a=256,\\:b=5\\sec^{6}\\left(θ\\right)\\tan^{4}\\left(θ\\right),\\:c=4\\tan^{6}\\left(θ\\right)\\sec^{4}\\left(θ\\right)$$" ], "result": "=256\\cdot\\:5\\sec^{6}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+256\\cdot\\:4\\tan^{6}\\left(θ\\right)\\sec^{4}\\left(θ\\right)", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "interim", "title": "Simplify $$256\\cdot\\:5\\sec^{6}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+256\\cdot\\:4\\tan^{6}\\left(θ\\right)\\sec^{4}\\left(θ\\right):{\\quad}1280\\sec^{6}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+1024\\tan^{6}\\left(θ\\right)\\sec^{4}\\left(θ\\right)$$", "input": "256\\cdot\\:5\\sec^{6}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+256\\cdot\\:4\\tan^{6}\\left(θ\\right)\\sec^{4}\\left(θ\\right)", "result": "=1280\\sec^{6}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+1024\\tan^{6}\\left(θ\\right)\\sec^{4}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$256\\cdot\\:5=1280$$", "result": "=1280\\sec^{6}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+256\\cdot\\:4\\tan^{6}\\left(θ\\right)\\sec^{4}\\left(θ\\right)" }, { "type": "step", "primary": "Multiply the numbers: $$256\\cdot\\:4=1024$$", "result": "=1280\\sec^{6}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+1024\\tan^{6}\\left(θ\\right)\\sec^{4}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7b5/2rR5O0jX5YmXhLqBXx1GGuUmWLwacJdjIdVxl7S50dEMe0BNAlN2YuniCZbL+V3W7ReUz4118M0GFIRK6+VXTSum/z5kLpMzXS1UJIey7fmnhl45Vhe9r4Acro3FI6Phv/FIvBdlBXMgJt7z7tXXVhSxc5xstaPVdv3Far+frlVdI2PhhZIyhe9ZwVoMldF8WBcH6+VFv8AcImSWkX3ql8XXPq6bNQlMm+36iNhkICJupZH8TkNKGCNAyF9lyYBKuFOHPlRCFLv3G+khHy0fMKl5qtbU7z9ksj5TiRTR2PqiE485aK+1WKPeIefwXSIUZGk+g6gqVwxD/XktSUw==" } }, { "type": "interim", "title": "Expand $$1048\\left(6\\sec^{6}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+3\\sec^{8}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right):{\\quad}6288\\sec^{6}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+3144\\sec^{8}\\left(θ\\right)\\tan^{2}\\left(θ\\right)$$", "input": "1048\\left(6\\sec^{6}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+3\\sec^{8}\\left(θ\\right)\\tan^{2}\\left(θ\\right)\\right)", "result": "=1280\\sec^{6}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+1024\\tan^{6}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+6288\\sec^{6}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+3144\\sec^{8}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+376\\left(8\\sec^{8}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{10}\\left(θ\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b+c\\right)=ab+ac$$", "secondary": [ "$$a=1048,\\:b=6\\sec^{6}\\left(θ\\right)\\tan^{4}\\left(θ\\right),\\:c=3\\sec^{8}\\left(θ\\right)\\tan^{2}\\left(θ\\right)$$" ], "result": "=1048\\cdot\\:6\\sec^{6}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+1048\\cdot\\:3\\sec^{8}\\left(θ\\right)\\tan^{2}\\left(θ\\right)", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "interim", "title": "Simplify $$1048\\cdot\\:6\\sec^{6}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+1048\\cdot\\:3\\sec^{8}\\left(θ\\right)\\tan^{2}\\left(θ\\right):{\\quad}6288\\sec^{6}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+3144\\sec^{8}\\left(θ\\right)\\tan^{2}\\left(θ\\right)$$", "input": "1048\\cdot\\:6\\sec^{6}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+1048\\cdot\\:3\\sec^{8}\\left(θ\\right)\\tan^{2}\\left(θ\\right)", "result": "=6288\\sec^{6}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+3144\\sec^{8}\\left(θ\\right)\\tan^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$1048\\cdot\\:6=6288$$", "result": "=6288\\sec^{6}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+1048\\cdot\\:3\\sec^{8}\\left(θ\\right)\\tan^{2}\\left(θ\\right)" }, { "type": "step", "primary": "Multiply the numbers: $$1048\\cdot\\:3=3144$$", "result": "=6288\\sec^{6}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+3144\\sec^{8}\\left(θ\\right)\\tan^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7OJe6Wsm1BFNjIHYVu5/v56FNDP4X2DkkvxYlQqF2ja18BpCWgpKemmZKH3M2PCMSOslpQ6aMQXwrZaWdgnUl9C061ljBSPJeENOw2efoSWsQkA/VmgnWKKgFLh4CGt4UamteA8UyY/eLm9jagP57hipgYsYO+tPxaijDqivmfqVcKydsXfw270bVn4Crg0LlSI6fe8VAe1xXU+QUS+zGf9bA+zX4bD3u3gx65o2NJhM0+d6MOE69Ua70IDOe7w/EamteA8UyY/eLm9jagP57hn3SEQIlzObJ/ycOecvQmyGNJQtnv9oWCSYdpSgqwSgLWDQ+pxqe8dF0wqkjjZjuMw==" } }, { "type": "interim", "title": "Expand $$376\\left(8\\sec^{8}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{10}\\left(θ\\right)\\right):{\\quad}3008\\sec^{8}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+376\\sec^{10}\\left(θ\\right)$$", "input": "376\\left(8\\sec^{8}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{10}\\left(θ\\right)\\right)", "result": "=1280\\sec^{6}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+1024\\tan^{6}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+6288\\sec^{6}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+3144\\sec^{8}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+3008\\sec^{8}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+376\\sec^{10}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b+c\\right)=ab+ac$$", "secondary": [ "$$a=376,\\:b=8\\sec^{8}\\left(θ\\right)\\tan^{2}\\left(θ\\right),\\:c=\\sec^{10}\\left(θ\\right)$$" ], "result": "=376\\cdot\\:8\\sec^{8}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+376\\sec^{10}\\left(θ\\right)", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "primary": "Multiply the numbers: $$376\\cdot\\:8=3008$$", "result": "=3008\\sec^{8}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+376\\sec^{10}\\left(θ\\right)" } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7XNVaZKNnv2s7DNmNSSmE45Lef8XrDaHRlN6SQVhckTDSGGfPWzLrxm/x6AXwcaSc0y8fJjnR2Fc8V+uNyaihSLjfCax7kNarsTfE38gXOXUyrHK5ao8I0Pk51+6LZK/LLio5x3kFEntFC67xOr2tzd4TI0WLQnrjwr8/06EeCM2jeh7+jKEzLb7VNCEMF3Z/uY/O5y3tloP7W2H6B2flSppVRuhFDJx9wN7DvhDK/UEFoGcgUavSD6OCtBw12AcjdnV7Rv1ZipL+mGGBWFtk/SS3daIZHtloJpe/PvtsyNI=" } }, { "type": "interim", "title": "Simplify $$1280\\sec^{6}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+1024\\tan^{6}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+6288\\sec^{6}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+3144\\sec^{8}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+3008\\sec^{8}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+376\\sec^{10}\\left(θ\\right):{\\quad}7568\\sec^{6}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+1024\\tan^{6}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+6152\\sec^{8}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+376\\sec^{10}\\left(θ\\right)$$", "input": "1280\\sec^{6}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+1024\\tan^{6}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+6288\\sec^{6}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+3144\\sec^{8}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+3008\\sec^{8}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+376\\sec^{10}\\left(θ\\right)", "result": "=7568\\sec^{6}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+1024\\tan^{6}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+6152\\sec^{8}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+376\\sec^{10}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Add similar elements: $$1280\\sec^{6}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+6288\\sec^{6}\\left(θ\\right)\\tan^{4}\\left(θ\\right)=7568\\sec^{6}\\left(θ\\right)\\tan^{4}\\left(θ\\right)$$", "result": "=7568\\sec^{6}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+1024\\tan^{6}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+3144\\sec^{8}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+3008\\sec^{8}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+376\\sec^{10}\\left(θ\\right)" }, { "type": "step", "primary": "Add similar elements: $$3144\\sec^{8}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+3008\\sec^{8}\\left(θ\\right)\\tan^{2}\\left(θ\\right)=6152\\sec^{8}\\left(θ\\right)\\tan^{2}\\left(θ\\right)$$", "result": "=7568\\sec^{6}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+1024\\tan^{6}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+6152\\sec^{8}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+376\\sec^{10}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=-8\\left(-4\\left(-4\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)+12\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)\\right)+12\\left(64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+132\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+14\\sec^{8}\\left(θ\\right)\\right)\\right)+24\\left(7568\\sec^{6}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+1024\\tan^{6}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+6152\\sec^{8}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+376\\sec^{10}\\left(θ\\right)\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=-8\\left(-4\\left(-4\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)+12\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)\\right)+12\\left(64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+132\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+14\\sec^{8}\\left(θ\\right)\\right)\\right)+24\\left(7568\\sec^{6}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+1024\\tan^{6}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+6152\\sec^{8}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+376\\sec^{10}\\left(θ\\right)\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "Evaluate $$-8\\left(-4\\left(-4\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)+12\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)\\right)+12\\left(64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+132\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+14\\sec^{8}\\left(θ\\right)\\right)\\right)+24\\left(7568\\sec^{6}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+1024\\tan^{6}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+6152\\sec^{8}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+376\\sec^{10}\\left(θ\\right)\\right)\\:$$at point $$θ=0:{\\quad}7936$$", "result": "=7936", "steps": [ { "type": "step", "primary": "Take the point $$θ=0$$ and plug it into $$-8\\left(-4\\left(-4\\left(-2\\sec^{2}\\left(θ\\right)+3\\sec^{4}\\left(θ\\right)\\right)+12\\left(4\\sec^{4}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+\\sec^{6}\\left(θ\\right)\\right)\\right)+12\\left(64\\sec^{4}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+132\\sec^{6}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+14\\sec^{8}\\left(θ\\right)\\right)\\right)+24\\left(7568\\sec^{6}\\left(θ\\right)\\tan^{4}\\left(θ\\right)+1024\\tan^{6}\\left(θ\\right)\\sec^{4}\\left(θ\\right)+6152\\sec^{8}\\left(θ\\right)\\tan^{2}\\left(θ\\right)+376\\sec^{10}\\left(θ\\right)\\right)$$", "result": "=-8\\left(-4\\left(-4\\left(-2\\sec^{2}\\left(0\\right)+3\\sec^{4}\\left(0\\right)\\right)+12\\left(4\\sec^{4}\\left(0\\right)\\tan^{2}\\left(0\\right)+\\sec^{6}\\left(0\\right)\\right)\\right)+12\\left(64\\sec^{4}\\left(0\\right)\\tan^{4}\\left(0\\right)+132\\sec^{6}\\left(0\\right)\\tan^{2}\\left(0\\right)+14\\sec^{8}\\left(0\\right)\\right)\\right)+24\\left(7568\\sec^{6}\\left(0\\right)\\tan^{4}\\left(0\\right)+1024\\tan^{6}\\left(0\\right)\\sec^{4}\\left(0\\right)+6152\\sec^{8}\\left(0\\right)\\tan^{2}\\left(0\\right)+376\\sec^{10}\\left(0\\right)\\right)" }, { "type": "interim", "title": "$$8\\left(-4\\left(-4\\left(-2\\sec^{2}\\left(0\\right)+3\\sec^{4}\\left(0\\right)\\right)+12\\left(4\\sec^{4}\\left(0\\right)\\tan^{2}\\left(0\\right)+\\sec^{6}\\left(0\\right)\\right)\\right)+12\\left(64\\sec^{4}\\left(0\\right)\\tan^{4}\\left(0\\right)+132\\sec^{6}\\left(0\\right)\\tan^{2}\\left(0\\right)+14\\sec^{8}\\left(0\\right)\\right)\\right)=1088$$", "input": "8\\left(-4\\left(-4\\left(-2\\sec^{2}\\left(0\\right)+3\\sec^{4}\\left(0\\right)\\right)+12\\left(4\\sec^{4}\\left(0\\right)\\tan^{2}\\left(0\\right)+\\sec^{6}\\left(0\\right)\\right)\\right)+12\\left(64\\sec^{4}\\left(0\\right)\\tan^{4}\\left(0\\right)+132\\sec^{6}\\left(0\\right)\\tan^{2}\\left(0\\right)+14\\sec^{8}\\left(0\\right)\\right)\\right)", "steps": [ { "type": "interim", "title": "$$4\\left(-4\\left(-2\\sec^{2}\\left(0\\right)+3\\sec^{4}\\left(0\\right)\\right)+12\\left(4\\sec^{4}\\left(0\\right)\\tan^{2}\\left(0\\right)+\\sec^{6}\\left(0\\right)\\right)\\right)=32$$", "input": "4\\left(-4\\left(-2\\sec^{2}\\left(0\\right)+3\\sec^{4}\\left(0\\right)\\right)+12\\left(4\\sec^{4}\\left(0\\right)\\tan^{2}\\left(0\\right)+\\sec^{6}\\left(0\\right)\\right)\\right)", "steps": [ { "type": "interim", "title": "$$4\\left(-2\\sec^{2}\\left(0\\right)+3\\sec^{4}\\left(0\\right)\\right)=4$$", "input": "4\\left(-2\\sec^{2}\\left(0\\right)+3\\sec^{4}\\left(0\\right)\\right)", "steps": [ { "type": "interim", "title": "$$2\\sec^{2}\\left(0\\right)=2$$", "input": "2\\sec^{2}\\left(0\\right)", "steps": [ { "type": "interim", "title": "$$\\sec^{2}\\left(0\\right)=1$$", "input": "\\sec^{2}\\left(0\\right)", "steps": [ { "type": "interim", "title": "Simplify $$\\sec\\left(0\\right):{\\quad}1$$", "input": "\\sec\\left(0\\right)", "result": "=1^{2}", "steps": [ { "type": "step", "primary": "Use the following trivial identity:$${\\quad}\\sec\\left(0\\right)=1$$", "secondary": [ "$$\\sec\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sec(x)&x&\\sec(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{2\\sqrt{3}}{3}&\\frac{7π}{6}&-\\frac{2\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&\\sqrt{2}&\\frac{5π}{4}&-\\sqrt{2}\\\\\\hline \\frac{π}{3}&2&\\frac{4π}{3}&-2\\\\\\hline \\frac{π}{2}&\\mathrm{Undefined}&\\frac{3π}{2}&\\mathrm{Undefined}\\\\\\hline \\frac{2π}{3}&-2&\\frac{5π}{3}&2\\\\\\hline \\frac{3π}{4}&-\\sqrt{2}&\\frac{7π}{4}&\\sqrt{2}\\\\\\hline \\frac{5π}{6}&-\\frac{2\\sqrt{3}}{3}&\\frac{11π}{6}&\\frac{2\\sqrt{3}}{3}\\\\\\hline \\end{array}$$" ], "result": "=1" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7y1xaZeQTSzp609voOcYk6VXTSum/z5kLpMzXS1UJIexiYFOaxxQqg3o8CNKkTm2qW+Sc57RV820557NJqaC71BfVjTo90KMWspnEdE4CKjE=" } }, { "type": "step", "result": "=2\\cdot\\:1" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=2" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7WiIIsb80YMnC/PcLPmbAry061ljBSPJeENOw2efoSWt8rNweSBKaNqhWM5iGGSVMdOMUzZRHPIlp1R8T2pkp20fQYskTwd8nVLo5risF1/I=" } }, { "type": "interim", "title": "$$3\\sec^{4}\\left(0\\right)=3$$", "input": "3\\sec^{4}\\left(0\\right)", "steps": [ { "type": "interim", "title": "$$\\sec^{4}\\left(0\\right)=1$$", "input": "\\sec^{4}\\left(0\\right)", "steps": [ { "type": "interim", "title": "Simplify $$\\sec\\left(0\\right):{\\quad}1$$", "input": "\\sec\\left(0\\right)", "result": "=1^{4}", "steps": [ { "type": "step", "primary": "Use the following trivial identity:$${\\quad}\\sec\\left(0\\right)=1$$", "secondary": [ "$$\\sec\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sec(x)&x&\\sec(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{2\\sqrt{3}}{3}&\\frac{7π}{6}&-\\frac{2\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&\\sqrt{2}&\\frac{5π}{4}&-\\sqrt{2}\\\\\\hline \\frac{π}{3}&2&\\frac{4π}{3}&-2\\\\\\hline \\frac{π}{2}&\\mathrm{Undefined}&\\frac{3π}{2}&\\mathrm{Undefined}\\\\\\hline \\frac{2π}{3}&-2&\\frac{5π}{3}&2\\\\\\hline \\frac{3π}{4}&-\\sqrt{2}&\\frac{7π}{4}&\\sqrt{2}\\\\\\hline \\frac{5π}{6}&-\\frac{2\\sqrt{3}}{3}&\\frac{11π}{6}&\\frac{2\\sqrt{3}}{3}\\\\\\hline \\end{array}$$" ], "result": "=1" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7TnbPBMsEd78kmmwfYDHqNVXTSum/z5kLpMzXS1UJIexiYFOaxxQqg3o8CNKkTm2qW+Sc57RV820557NJqaC71Ka6Ma3rg33eDIhfX/iuVnM=" } }, { "type": "step", "result": "=3\\cdot\\:1" }, { "type": "step", "primary": "Multiply the numbers: $$3\\cdot\\:1=3$$", "result": "=3" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/DCWessgznXvbIt+KUH6Sy061ljBSPJeENOw2efoSWuWTYGorfWzSEgF1o3crWNtABnhaHvVHSBGAVNgbL75+DAm81Br6rrrU7LVgwPHPmM=" } }, { "type": "step", "result": "=4\\left(3-2\\right)" }, { "type": "step", "primary": "Add/Subtract the numbers: $$-2+3=1$$", "result": "=4\\cdot\\:1" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:1=4$$", "result": "=4" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7SGVSB79Ogsx4oIuNaHoM5zOWn8zcWOZXWMkYneaKW/PTLx8mOdHYVzxX643JqKFIlFYe8lM/WYPz8s80IajfKB0R3c0uYNhyRWpx951US4jV06frS5K/+G+6QAS0YeGxwllEClEEu5y+6CgQm6mC37CI2sSeA74029n2yo277ZU=" } }, { "type": "interim", "title": "$$12\\left(4\\sec^{4}\\left(0\\right)\\tan^{2}\\left(0\\right)+\\sec^{6}\\left(0\\right)\\right)=12$$", "input": "12\\left(4\\sec^{4}\\left(0\\right)\\tan^{2}\\left(0\\right)+\\sec^{6}\\left(0\\right)\\right)", "steps": [ { "type": "interim", "title": "$$4\\sec^{4}\\left(0\\right)\\tan^{2}\\left(0\\right)=0$$", "input": "4\\sec^{4}\\left(0\\right)\\tan^{2}\\left(0\\right)", "steps": [ { "type": "interim", "title": "$$\\sec^{4}\\left(0\\right)=1$$", "input": "\\sec^{4}\\left(0\\right)", "steps": [ { "type": "interim", "title": "Simplify $$\\sec\\left(0\\right):{\\quad}1$$", "input": "\\sec\\left(0\\right)", "result": "=1^{4}", "steps": [ { "type": "step", "primary": "Use the following trivial identity:$${\\quad}\\sec\\left(0\\right)=1$$", "secondary": [ "$$\\sec\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sec(x)&x&\\sec(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{2\\sqrt{3}}{3}&\\frac{7π}{6}&-\\frac{2\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&\\sqrt{2}&\\frac{5π}{4}&-\\sqrt{2}\\\\\\hline \\frac{π}{3}&2&\\frac{4π}{3}&-2\\\\\\hline \\frac{π}{2}&\\mathrm{Undefined}&\\frac{3π}{2}&\\mathrm{Undefined}\\\\\\hline \\frac{2π}{3}&-2&\\frac{5π}{3}&2\\\\\\hline \\frac{3π}{4}&-\\sqrt{2}&\\frac{7π}{4}&\\sqrt{2}\\\\\\hline \\frac{5π}{6}&-\\frac{2\\sqrt{3}}{3}&\\frac{11π}{6}&\\frac{2\\sqrt{3}}{3}\\\\\\hline \\end{array}$$" ], "result": "=1" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7TnbPBMsEd78kmmwfYDHqNVXTSum/z5kLpMzXS1UJIexiYFOaxxQqg3o8CNKkTm2qW+Sc57RV820557NJqaC71Ka6Ma3rg33eDIhfX/iuVnM=" } }, { "type": "step", "result": "=4\\cdot\\:1\\cdot\\:\\tan^{2}\\left(0\\right)" }, { "type": "interim", "title": "$$\\tan^{2}\\left(0\\right)=0$$", "input": "\\tan^{2}\\left(0\\right)", "steps": [ { "type": "interim", "title": "Simplify $$\\tan\\left(0\\right):{\\quad}0$$", "input": "\\tan\\left(0\\right)", "result": "=0^{2}", "steps": [ { "type": "step", "primary": "Use the following trivial identity:$${\\quad}\\tan\\left(0\\right)=0$$", "secondary": [ "$$\\tan\\left(x\\right)$$ periodicity table with $$πn$$ cycle:<br/>$$\\begin{array}{|c|c|}\\hline x&\\tan(x)\\\\\\hline 0&0\\\\\\hline \\frac{π}{6}&\\frac{\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&1\\\\\\hline \\frac{π}{3}&\\sqrt{3}\\\\\\hline \\frac{π}{2}&\\pm\\infty\\\\\\hline \\frac{2π}{3}&-\\sqrt{3}\\\\\\hline \\frac{3π}{4}&-1\\\\\\hline \\frac{5π}{6}&-\\frac{\\sqrt{3}}{3}\\\\\\hline &\\\\\\hline \\end{array}$$" ], "result": "=0" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Apply rule $$0^{a}=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7EFsToMX1NPjPuXVSALucCFXTSum/z5kLpMzXS1UJIeyqjfZqhkrOoAz8sRb7wXZWPGHkI0MFUiTEwg/g9Re5WBSGyzDy0veA23V3wI6HDZ0=" } }, { "type": "step", "result": "=4\\cdot\\:1\\cdot\\:0" }, { "type": "step", "primary": "Apply rule $$0\\cdot\\:a=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s78lx/rXvIfCfIw4X8Nv9JwGpb6Zt5uD6Xe9ICIhSFAqbdd47a0hQ8flDbGsI5To1dTbAOxT8wOTlsw5gGf+Hdr8r4WVs4CtopdU5PWLTw1eRWrxRA8eaaGXYlDOSEM378ENKOm3B8+xM3yGSItlc9gA==" } }, { "type": "interim", "title": "$$\\sec^{6}\\left(0\\right)=1$$", "input": "\\sec^{6}\\left(0\\right)", "steps": [ { "type": "interim", "title": "Simplify $$\\sec\\left(0\\right):{\\quad}1$$", "input": "\\sec\\left(0\\right)", "result": "=1^{6}", "steps": [ { "type": "step", "primary": "Use the following trivial identity:$${\\quad}\\sec\\left(0\\right)=1$$", "secondary": [ "$$\\sec\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sec(x)&x&\\sec(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{2\\sqrt{3}}{3}&\\frac{7π}{6}&-\\frac{2\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&\\sqrt{2}&\\frac{5π}{4}&-\\sqrt{2}\\\\\\hline \\frac{π}{3}&2&\\frac{4π}{3}&-2\\\\\\hline \\frac{π}{2}&\\mathrm{Undefined}&\\frac{3π}{2}&\\mathrm{Undefined}\\\\\\hline \\frac{2π}{3}&-2&\\frac{5π}{3}&2\\\\\\hline \\frac{3π}{4}&-\\sqrt{2}&\\frac{7π}{4}&\\sqrt{2}\\\\\\hline \\frac{5π}{6}&-\\frac{2\\sqrt{3}}{3}&\\frac{11π}{6}&\\frac{2\\sqrt{3}}{3}\\\\\\hline \\end{array}$$" ], "result": "=1" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7WcSYqbVPn/qsbXXyqHdMbFXTSum/z5kLpMzXS1UJIexiYFOaxxQqg3o8CNKkTm2qW+Sc57RV820557NJqaC71Idwe88IvH72vjqcga7gyBg=" } }, { "type": "step", "result": "=12\\left(0+1\\right)" }, { "type": "step", "primary": "Add the numbers: $$0+1=1$$", "result": "=12\\cdot\\:1" }, { "type": "step", "primary": "Multiply the numbers: $$12\\cdot\\:1=12$$", "result": "=12" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7lv52tOJZUIL6vqIybdOpTv6k/0Ej05jaZlvKuZ7BchqkNRU526e9DUedYVjkPVRIICf2WQN9mSJxQaQ7cQX4iveB/AqBp60vexPOzIrQ2kDh6XJOU+ADur+UsyVO4as2WfKjICzStUBkrvQwB+qiDavP5dLtToEw2Zpkikf00/ZOjeRcPaACRL5La6ZwWTik" } }, { "type": "step", "result": "=4\\left(12-4\\right)" }, { "type": "step", "primary": "Add/Subtract the numbers: $$-4+12=8$$", "result": "=4\\cdot\\:8" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:8=32$$", "result": "=32" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7gnkt+qYVJeyRH8AcpPA53w9zA7z3N/kwupn+iB41A+xqlU3QAFTFK+yGmPKWoDuzazhA0+a1wf9G1fiOsZuQNzomUw5kYyCrSM1rC9ftEoUAlilG71elit3w1IBbYN0POTgB3b1cYqKZAW4Ah/4Vox0R3c0uYNhyRWpx951US4jYY6TiTc30dYTBcnzGf4+8M5afzNxY5ldYyRid5opb84FAqMTCfEIxPAs+HdURhAz+pP9BI9OY2mZbyrmewXIaFrzMspXNVTtj+lSEWrtmDbCI2sSeA74029n2yo277ZU=" } }, { "type": "interim", "title": "$$12\\left(64\\sec^{4}\\left(0\\right)\\tan^{4}\\left(0\\right)+132\\sec^{6}\\left(0\\right)\\tan^{2}\\left(0\\right)+14\\sec^{8}\\left(0\\right)\\right)=168$$", "input": "12\\left(64\\sec^{4}\\left(0\\right)\\tan^{4}\\left(0\\right)+132\\sec^{6}\\left(0\\right)\\tan^{2}\\left(0\\right)+14\\sec^{8}\\left(0\\right)\\right)", "steps": [ { "type": "interim", "title": "$$64\\sec^{4}\\left(0\\right)\\tan^{4}\\left(0\\right)=0$$", "input": "64\\sec^{4}\\left(0\\right)\\tan^{4}\\left(0\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{m}b^{m}=\\left(ab\\right)^{m}$$", "secondary": [ "$$\\sec^{4}\\left(0\\right)\\tan^{4}\\left(0\\right)=\\left(\\sec\\left(0\\right)\\tan\\left(0\\right)\\right)^{4}$$" ], "result": "=64\\left(\\sec\\left(0\\right)\\tan\\left(0\\right)\\right)^{4}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "$$\\sec\\left(0\\right)\\tan\\left(0\\right)=0$$", "input": "\\sec\\left(0\\right)\\tan\\left(0\\right)", "steps": [ { "type": "interim", "title": "Simplify $$\\sec\\left(0\\right):{\\quad}1$$", "input": "\\sec\\left(0\\right)", "result": "=1\\cdot\\:\\tan\\left(0\\right)", "steps": [ { "type": "step", "primary": "Use the following trivial identity:$${\\quad}\\sec\\left(0\\right)=1$$", "secondary": [ "$$\\sec\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sec(x)&x&\\sec(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{2\\sqrt{3}}{3}&\\frac{7π}{6}&-\\frac{2\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&\\sqrt{2}&\\frac{5π}{4}&-\\sqrt{2}\\\\\\hline \\frac{π}{3}&2&\\frac{4π}{3}&-2\\\\\\hline \\frac{π}{2}&\\mathrm{Undefined}&\\frac{3π}{2}&\\mathrm{Undefined}\\\\\\hline \\frac{2π}{3}&-2&\\frac{5π}{3}&2\\\\\\hline \\frac{3π}{4}&-\\sqrt{2}&\\frac{7π}{4}&\\sqrt{2}\\\\\\hline \\frac{5π}{6}&-\\frac{2\\sqrt{3}}{3}&\\frac{11π}{6}&\\frac{2\\sqrt{3}}{3}\\\\\\hline \\end{array}$$" ], "result": "=1" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "interim", "title": "Simplify $$\\tan\\left(0\\right):{\\quad}0$$", "input": "\\tan\\left(0\\right)", "result": "=1\\cdot\\:0", "steps": [ { "type": "step", "primary": "Use the following trivial identity:$${\\quad}\\tan\\left(0\\right)=0$$", "secondary": [ "$$\\tan\\left(x\\right)$$ periodicity table with $$πn$$ cycle:<br/>$$\\begin{array}{|c|c|}\\hline x&\\tan(x)\\\\\\hline 0&0\\\\\\hline \\frac{π}{6}&\\frac{\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&1\\\\\\hline \\frac{π}{3}&\\sqrt{3}\\\\\\hline \\frac{π}{2}&\\pm\\infty\\\\\\hline \\frac{2π}{3}&-\\sqrt{3}\\\\\\hline \\frac{3π}{4}&-1\\\\\\hline \\frac{5π}{6}&-\\frac{\\sqrt{3}}{3}\\\\\\hline &\\\\\\hline \\end{array}$$" ], "result": "=0" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Apply rule $$0\\cdot\\:a=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7wZhLm0FifI5RV/wdUY/V+xs4UrvXrnDv0XDlfU7q7ckE5aqGN/sLZfeoFZRwtGLqP8vQyhiD4JSfqjIvcQ7tinNBgJ91kDwEQa4/+Opm+A8QRYlc0DxGPn21AQgIWYWg" } }, { "type": "step", "result": "=0^{4}\\cdot\\:64" }, { "type": "step", "primary": "Apply rule $$0^{a}=0$$", "result": "=64\\cdot\\:0" }, { "type": "step", "primary": "Apply rule $$0\\cdot\\:a=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7jCpfsFjEU27xNecKdQGhyN+vnf11FNHf3iNt9M+J8Z8gJ/ZZA32ZInFBpDtxBfiKf7LqB9CcyvYCWDsGseX09kk96RFLStITDoCO0hmLjEzqP13v7CCUzv+TjUxfUDRqCFX7kPfEkIuYfqIFW07ybg==" } }, { "type": "interim", "title": "$$132\\sec^{6}\\left(0\\right)\\tan^{2}\\left(0\\right)=0$$", "input": "132\\sec^{6}\\left(0\\right)\\tan^{2}\\left(0\\right)", "steps": [ { "type": "interim", "title": "$$\\sec^{6}\\left(0\\right)=1$$", "input": "\\sec^{6}\\left(0\\right)", "steps": [ { "type": "interim", "title": "Simplify $$\\sec\\left(0\\right):{\\quad}1$$", "input": "\\sec\\left(0\\right)", "result": "=1^{6}", "steps": [ { "type": "step", "primary": "Use the following trivial identity:$${\\quad}\\sec\\left(0\\right)=1$$", "secondary": [ "$$\\sec\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sec(x)&x&\\sec(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{2\\sqrt{3}}{3}&\\frac{7π}{6}&-\\frac{2\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&\\sqrt{2}&\\frac{5π}{4}&-\\sqrt{2}\\\\\\hline \\frac{π}{3}&2&\\frac{4π}{3}&-2\\\\\\hline \\frac{π}{2}&\\mathrm{Undefined}&\\frac{3π}{2}&\\mathrm{Undefined}\\\\\\hline \\frac{2π}{3}&-2&\\frac{5π}{3}&2\\\\\\hline \\frac{3π}{4}&-\\sqrt{2}&\\frac{7π}{4}&\\sqrt{2}\\\\\\hline \\frac{5π}{6}&-\\frac{2\\sqrt{3}}{3}&\\frac{11π}{6}&\\frac{2\\sqrt{3}}{3}\\\\\\hline \\end{array}$$" ], "result": "=1" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7WcSYqbVPn/qsbXXyqHdMbFXTSum/z5kLpMzXS1UJIexiYFOaxxQqg3o8CNKkTm2qW+Sc57RV820557NJqaC71Idwe88IvH72vjqcga7gyBg=" } }, { "type": "step", "result": "=132\\cdot\\:1\\cdot\\:\\tan^{2}\\left(0\\right)" }, { "type": "interim", "title": "$$\\tan^{2}\\left(0\\right)=0$$", "input": "\\tan^{2}\\left(0\\right)", "steps": [ { "type": "interim", "title": "Simplify $$\\tan\\left(0\\right):{\\quad}0$$", "input": "\\tan\\left(0\\right)", "result": "=0^{2}", "steps": [ { "type": "step", "primary": "Use the following trivial identity:$${\\quad}\\tan\\left(0\\right)=0$$", "secondary": [ "$$\\tan\\left(x\\right)$$ periodicity table with $$πn$$ cycle:<br/>$$\\begin{array}{|c|c|}\\hline x&\\tan(x)\\\\\\hline 0&0\\\\\\hline \\frac{π}{6}&\\frac{\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&1\\\\\\hline \\frac{π}{3}&\\sqrt{3}\\\\\\hline \\frac{π}{2}&\\pm\\infty\\\\\\hline \\frac{2π}{3}&-\\sqrt{3}\\\\\\hline \\frac{3π}{4}&-1\\\\\\hline \\frac{5π}{6}&-\\frac{\\sqrt{3}}{3}\\\\\\hline &\\\\\\hline \\end{array}$$" ], "result": "=0" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Apply rule $$0^{a}=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7EFsToMX1NPjPuXVSALucCFXTSum/z5kLpMzXS1UJIeyqjfZqhkrOoAz8sRb7wXZWPGHkI0MFUiTEwg/g9Re5WBSGyzDy0veA23V3wI6HDZ0=" } }, { "type": "step", "result": "=132\\cdot\\:1\\cdot\\:0" }, { "type": "step", "primary": "Apply rule $$0\\cdot\\:a=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7hDHcMpcibER9XzN1Yg7CFW7J22bskQyWoBdfBnHASMvehkKrn0era9rz8TlL+x/vZuJKdCFsPJy1+5gBMEc9dsdkCcWWh42PXyyVg0WHsjHh/R/PNQmbxXqoSPkeiOuEYAQCAfHsRO0+u9TOmQBHnA==" } }, { "type": "interim", "title": "$$14\\sec^{8}\\left(0\\right)=14$$", "input": "14\\sec^{8}\\left(0\\right)", "steps": [ { "type": "interim", "title": "$$\\sec^{8}\\left(0\\right)=1$$", "input": "\\sec^{8}\\left(0\\right)", "steps": [ { "type": "interim", "title": "Simplify $$\\sec\\left(0\\right):{\\quad}1$$", "input": "\\sec\\left(0\\right)", "result": "=1^{8}", "steps": [ { "type": "step", "primary": "Use the following trivial identity:$${\\quad}\\sec\\left(0\\right)=1$$", "secondary": [ "$$\\sec\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sec(x)&x&\\sec(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{2\\sqrt{3}}{3}&\\frac{7π}{6}&-\\frac{2\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&\\sqrt{2}&\\frac{5π}{4}&-\\sqrt{2}\\\\\\hline \\frac{π}{3}&2&\\frac{4π}{3}&-2\\\\\\hline \\frac{π}{2}&\\mathrm{Undefined}&\\frac{3π}{2}&\\mathrm{Undefined}\\\\\\hline \\frac{2π}{3}&-2&\\frac{5π}{3}&2\\\\\\hline \\frac{3π}{4}&-\\sqrt{2}&\\frac{7π}{4}&\\sqrt{2}\\\\\\hline \\frac{5π}{6}&-\\frac{2\\sqrt{3}}{3}&\\frac{11π}{6}&\\frac{2\\sqrt{3}}{3}\\\\\\hline \\end{array}$$" ], "result": "=1" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7p6IxEfKejassVlEaaYgXqFXTSum/z5kLpMzXS1UJIexiYFOaxxQqg3o8CNKkTm2qW+Sc57RV820557NJqaC71Dd1z7LYeV9QGyBQ1bCNf14=" } }, { "type": "step", "result": "=14\\cdot\\:1" }, { "type": "step", "primary": "Multiply the numbers: $$14\\cdot\\:1=14$$", "result": "=14" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7CKfnUOCj05zZQSfHGtbxZgCWKUbvV6WK3fDUgFtg3Q9I5fbYu61vDlo8ZYCzwCZ5jFF+Grhte/2UqFkzsPs4p48kfsp1NuZUpmGmGCq3EH6wiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=12\\left(0+0+14\\right)" }, { "type": "step", "primary": "Add the numbers: $$0+0+14=14$$", "result": "=12\\cdot\\:14" }, { "type": "step", "primary": "Multiply the numbers: $$12\\cdot\\:14=168$$", "result": "=168" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7g9TcsEzV7InqwuZOLGfZt1xgK+yPINzyLi5gcEB0VbiOfWofmcg617UzDkH+2fVbyR3NdNPppln3fSgAjl+TBB1Pl0yjHH89T351Qec2oqHdd47a0hQ8flDbGsI5To1d6bCj/TT6XHxrDtqe3I/moYuU7v8WgEXr+cueuwDK5+0zq+QS9kblohPScFnBjh/0CTAFHH16SB6RSMGSVmr0OZWBls1W2NWhPslgpbulGoy5I4rnDsAX1ck0KsGYmrx3njE6qal773yXHTzA9rZZSQ==" } }, { "type": "step", "result": "=8\\left(168-32\\right)" }, { "type": "step", "primary": "Add/Subtract the numbers: $$-32+168=136$$", "result": "=8\\cdot\\:136" }, { "type": "step", "primary": "Multiply the numbers: $$8\\cdot\\:136=1088$$", "result": "=1088" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "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" } }, { "type": "interim", "title": "$$24\\left(7568\\sec^{6}\\left(0\\right)\\tan^{4}\\left(0\\right)+1024\\tan^{6}\\left(0\\right)\\sec^{4}\\left(0\\right)+6152\\sec^{8}\\left(0\\right)\\tan^{2}\\left(0\\right)+376\\sec^{10}\\left(0\\right)\\right)=9024$$", "input": "24\\left(7568\\sec^{6}\\left(0\\right)\\tan^{4}\\left(0\\right)+1024\\tan^{6}\\left(0\\right)\\sec^{4}\\left(0\\right)+6152\\sec^{8}\\left(0\\right)\\tan^{2}\\left(0\\right)+376\\sec^{10}\\left(0\\right)\\right)", "steps": [ { "type": "interim", "title": "$$7568\\sec^{6}\\left(0\\right)\\tan^{4}\\left(0\\right)=0$$", "input": "7568\\sec^{6}\\left(0\\right)\\tan^{4}\\left(0\\right)", "steps": [ { "type": "interim", "title": "$$\\sec^{6}\\left(0\\right)=1$$", "input": "\\sec^{6}\\left(0\\right)", "steps": [ { "type": "interim", "title": "Simplify $$\\sec\\left(0\\right):{\\quad}1$$", "input": "\\sec\\left(0\\right)", "result": "=1^{6}", "steps": [ { "type": "step", "primary": "Use the following trivial identity:$${\\quad}\\sec\\left(0\\right)=1$$", "secondary": [ "$$\\sec\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sec(x)&x&\\sec(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{2\\sqrt{3}}{3}&\\frac{7π}{6}&-\\frac{2\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&\\sqrt{2}&\\frac{5π}{4}&-\\sqrt{2}\\\\\\hline \\frac{π}{3}&2&\\frac{4π}{3}&-2\\\\\\hline \\frac{π}{2}&\\mathrm{Undefined}&\\frac{3π}{2}&\\mathrm{Undefined}\\\\\\hline \\frac{2π}{3}&-2&\\frac{5π}{3}&2\\\\\\hline \\frac{3π}{4}&-\\sqrt{2}&\\frac{7π}{4}&\\sqrt{2}\\\\\\hline \\frac{5π}{6}&-\\frac{2\\sqrt{3}}{3}&\\frac{11π}{6}&\\frac{2\\sqrt{3}}{3}\\\\\\hline \\end{array}$$" ], "result": "=1" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7WcSYqbVPn/qsbXXyqHdMbFXTSum/z5kLpMzXS1UJIexiYFOaxxQqg3o8CNKkTm2qW+Sc57RV820557NJqaC71Idwe88IvH72vjqcga7gyBg=" } }, { "type": "step", "result": "=7568\\cdot\\:1\\cdot\\:\\tan^{4}\\left(0\\right)" }, { "type": "interim", "title": "$$\\tan^{4}\\left(0\\right)=0$$", "input": "\\tan^{4}\\left(0\\right)", "steps": [ { "type": "interim", "title": "Simplify $$\\tan\\left(0\\right):{\\quad}0$$", "input": "\\tan\\left(0\\right)", "result": "=0^{4}", "steps": [ { "type": "step", "primary": "Use the following trivial identity:$${\\quad}\\tan\\left(0\\right)=0$$", "secondary": [ "$$\\tan\\left(x\\right)$$ periodicity table with $$πn$$ cycle:<br/>$$\\begin{array}{|c|c|}\\hline x&\\tan(x)\\\\\\hline 0&0\\\\\\hline \\frac{π}{6}&\\frac{\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&1\\\\\\hline \\frac{π}{3}&\\sqrt{3}\\\\\\hline \\frac{π}{2}&\\pm\\infty\\\\\\hline \\frac{2π}{3}&-\\sqrt{3}\\\\\\hline \\frac{3π}{4}&-1\\\\\\hline \\frac{5π}{6}&-\\frac{\\sqrt{3}}{3}\\\\\\hline &\\\\\\hline \\end{array}$$" ], "result": "=0" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Apply rule $$0^{a}=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7AhMdrQkztAm4Sd9xEOfLM1XTSum/z5kLpMzXS1UJIeyqjfZqhkrOoAz8sRb7wXZWPGHkI0MFUiTEwg/g9Re5WEAUWA/HI7+fZNhNFJm5Ep8=" } }, { "type": "step", "result": "=7568\\cdot\\:1\\cdot\\:0" }, { "type": "step", "primary": "Apply rule $$0\\cdot\\:a=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s76BEF/nT13bGWEXUP9thi494TJAF0Inm0N6e7prGY/11V00rpv8+ZC6TM10tVCSHsqo32aoZKzqAM/LEW+8F2VkNNJtD9uB9LkdF+tnZZtwJZXo5elHJQ7S0xAMPEX14gQBRYD8cjv59k2E0UmbkSnw==" } }, { "type": "interim", "title": "$$1024\\tan^{6}\\left(0\\right)\\sec^{4}\\left(0\\right)=0$$", "input": "1024\\tan^{6}\\left(0\\right)\\sec^{4}\\left(0\\right)", "steps": [ { "type": "interim", "title": "$$\\tan^{6}\\left(0\\right)=0$$", "input": "\\tan^{6}\\left(0\\right)", "steps": [ { "type": "interim", "title": "Simplify $$\\tan\\left(0\\right):{\\quad}0$$", "input": "\\tan\\left(0\\right)", "result": "=0^{6}", "steps": [ { "type": "step", "primary": "Use the following trivial identity:$${\\quad}\\tan\\left(0\\right)=0$$", "secondary": [ "$$\\tan\\left(x\\right)$$ periodicity table with $$πn$$ cycle:<br/>$$\\begin{array}{|c|c|}\\hline x&\\tan(x)\\\\\\hline 0&0\\\\\\hline \\frac{π}{6}&\\frac{\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&1\\\\\\hline \\frac{π}{3}&\\sqrt{3}\\\\\\hline \\frac{π}{2}&\\pm\\infty\\\\\\hline \\frac{2π}{3}&-\\sqrt{3}\\\\\\hline \\frac{3π}{4}&-1\\\\\\hline \\frac{5π}{6}&-\\frac{\\sqrt{3}}{3}\\\\\\hline &\\\\\\hline \\end{array}$$" ], "result": "=0" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Apply rule $$0^{a}=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7MGf/qfyYiohrfrJJ4jgFPFXTSum/z5kLpMzXS1UJIeyqjfZqhkrOoAz8sRb7wXZWPGHkI0MFUiTEwg/g9Re5WMuHCSmLhTnL5XpqgJXecYo=" } }, { "type": "step", "result": "=1024\\cdot\\:0\\cdot\\:\\sec^{4}\\left(0\\right)" }, { "type": "interim", "title": "$$\\sec^{4}\\left(0\\right)=1$$", "input": "\\sec^{4}\\left(0\\right)", "steps": [ { "type": "interim", "title": "Simplify $$\\sec\\left(0\\right):{\\quad}1$$", "input": "\\sec\\left(0\\right)", "result": "=1^{4}", "steps": [ { "type": "step", "primary": "Use the following trivial identity:$${\\quad}\\sec\\left(0\\right)=1$$", "secondary": [ "$$\\sec\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sec(x)&x&\\sec(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{2\\sqrt{3}}{3}&\\frac{7π}{6}&-\\frac{2\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&\\sqrt{2}&\\frac{5π}{4}&-\\sqrt{2}\\\\\\hline \\frac{π}{3}&2&\\frac{4π}{3}&-2\\\\\\hline \\frac{π}{2}&\\mathrm{Undefined}&\\frac{3π}{2}&\\mathrm{Undefined}\\\\\\hline \\frac{2π}{3}&-2&\\frac{5π}{3}&2\\\\\\hline \\frac{3π}{4}&-\\sqrt{2}&\\frac{7π}{4}&\\sqrt{2}\\\\\\hline \\frac{5π}{6}&-\\frac{2\\sqrt{3}}{3}&\\frac{11π}{6}&\\frac{2\\sqrt{3}}{3}\\\\\\hline \\end{array}$$" ], "result": "=1" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7TnbPBMsEd78kmmwfYDHqNVXTSum/z5kLpMzXS1UJIexiYFOaxxQqg3o8CNKkTm2qW+Sc57RV820557NJqaC71Ka6Ma3rg33eDIhfX/iuVnM=" } }, { "type": "step", "result": "=1024\\cdot\\:0\\cdot\\:1" }, { "type": "step", "primary": "Apply rule $$0\\cdot\\:a=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7sGESuCr6vFRH/dePfLbt9GUI8kNrW0+dmZq7KLHLnplV00rpv8+ZC6TM10tVCSHsqo32aoZKzqAM/LEW+8F2VsPi6Q20l8NW6eXsUOxRORMj9x/OPTVmUqZbpkbMYvTJWliDQHwx5daoL0Fqu6ANfg==" } }, { "type": "interim", "title": "$$6152\\sec^{8}\\left(0\\right)\\tan^{2}\\left(0\\right)=0$$", "input": "6152\\sec^{8}\\left(0\\right)\\tan^{2}\\left(0\\right)", "steps": [ { "type": "interim", "title": "$$\\sec^{8}\\left(0\\right)=1$$", "input": "\\sec^{8}\\left(0\\right)", "steps": [ { "type": "interim", "title": "Simplify $$\\sec\\left(0\\right):{\\quad}1$$", "input": "\\sec\\left(0\\right)", "result": "=1^{8}", "steps": [ { "type": "step", "primary": "Use the following trivial identity:$${\\quad}\\sec\\left(0\\right)=1$$", "secondary": [ "$$\\sec\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sec(x)&x&\\sec(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{2\\sqrt{3}}{3}&\\frac{7π}{6}&-\\frac{2\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&\\sqrt{2}&\\frac{5π}{4}&-\\sqrt{2}\\\\\\hline \\frac{π}{3}&2&\\frac{4π}{3}&-2\\\\\\hline \\frac{π}{2}&\\mathrm{Undefined}&\\frac{3π}{2}&\\mathrm{Undefined}\\\\\\hline \\frac{2π}{3}&-2&\\frac{5π}{3}&2\\\\\\hline \\frac{3π}{4}&-\\sqrt{2}&\\frac{7π}{4}&\\sqrt{2}\\\\\\hline \\frac{5π}{6}&-\\frac{2\\sqrt{3}}{3}&\\frac{11π}{6}&\\frac{2\\sqrt{3}}{3}\\\\\\hline \\end{array}$$" ], "result": "=1" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7p6IxEfKejassVlEaaYgXqFXTSum/z5kLpMzXS1UJIexiYFOaxxQqg3o8CNKkTm2qW+Sc57RV820557NJqaC71Dd1z7LYeV9QGyBQ1bCNf14=" } }, { "type": "step", "result": "=6152\\cdot\\:1\\cdot\\:\\tan^{2}\\left(0\\right)" }, { "type": "interim", "title": "$$\\tan^{2}\\left(0\\right)=0$$", "input": "\\tan^{2}\\left(0\\right)", "steps": [ { "type": "interim", "title": "Simplify $$\\tan\\left(0\\right):{\\quad}0$$", "input": "\\tan\\left(0\\right)", "result": "=0^{2}", "steps": [ { "type": "step", "primary": "Use the following trivial identity:$${\\quad}\\tan\\left(0\\right)=0$$", "secondary": [ "$$\\tan\\left(x\\right)$$ periodicity table with $$πn$$ cycle:<br/>$$\\begin{array}{|c|c|}\\hline x&\\tan(x)\\\\\\hline 0&0\\\\\\hline \\frac{π}{6}&\\frac{\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&1\\\\\\hline \\frac{π}{3}&\\sqrt{3}\\\\\\hline \\frac{π}{2}&\\pm\\infty\\\\\\hline \\frac{2π}{3}&-\\sqrt{3}\\\\\\hline \\frac{3π}{4}&-1\\\\\\hline \\frac{5π}{6}&-\\frac{\\sqrt{3}}{3}\\\\\\hline &\\\\\\hline \\end{array}$$" ], "result": "=0" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Apply rule $$0^{a}=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7EFsToMX1NPjPuXVSALucCFXTSum/z5kLpMzXS1UJIeyqjfZqhkrOoAz8sRb7wXZWPGHkI0MFUiTEwg/g9Re5WBSGyzDy0veA23V3wI6HDZ0=" } }, { "type": "step", "result": "=6152\\cdot\\:1\\cdot\\:0" }, { "type": "step", "primary": "Apply rule $$0\\cdot\\:a=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7MVsVv1+CS4743pLneI/fZ/76RFCatHoTFVB0uJUqMtdV00rpv8+ZC6TM10tVCSHsqo32aoZKzqAM/LEW+8F2Vk6/GfEbEmy4ZJCK9k4wN2cVoZzu237DbC8FCGLi6LGGFIbLMPLS94DbdXfAjocNnQ==" } }, { "type": "interim", "title": "$$376\\sec^{10}\\left(0\\right)=376$$", "input": "376\\sec^{10}\\left(0\\right)", "steps": [ { "type": "interim", "title": "$$\\sec^{10}\\left(0\\right)=1$$", "input": "\\sec^{10}\\left(0\\right)", "steps": [ { "type": "interim", "title": "Simplify $$\\sec\\left(0\\right):{\\quad}1$$", "input": "\\sec\\left(0\\right)", "result": "=1^{10}", "steps": [ { "type": "step", "primary": "Use the following trivial identity:$${\\quad}\\sec\\left(0\\right)=1$$", "secondary": [ "$$\\sec\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sec(x)&x&\\sec(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{2\\sqrt{3}}{3}&\\frac{7π}{6}&-\\frac{2\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&\\sqrt{2}&\\frac{5π}{4}&-\\sqrt{2}\\\\\\hline \\frac{π}{3}&2&\\frac{4π}{3}&-2\\\\\\hline \\frac{π}{2}&\\mathrm{Undefined}&\\frac{3π}{2}&\\mathrm{Undefined}\\\\\\hline \\frac{2π}{3}&-2&\\frac{5π}{3}&2\\\\\\hline \\frac{3π}{4}&-\\sqrt{2}&\\frac{7π}{4}&\\sqrt{2}\\\\\\hline \\frac{5π}{6}&-\\frac{2\\sqrt{3}}{3}&\\frac{11π}{6}&\\frac{2\\sqrt{3}}{3}\\\\\\hline \\end{array}$$" ], "result": "=1" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7MzqAL9EhfQY+3jHTWw41Gi061ljBSPJeENOw2efoSWs8auWUd4WNoosLPjGkhvjR9p7cPOxUNnMSMAsqahfW4iUfRG8ONZLjJIf/Hw0tMxU=" } }, { "type": "step", "result": "=376\\cdot\\:1" }, { "type": "step", "primary": "Multiply the numbers: $$376\\cdot\\:1=376$$", "result": "=376" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7c3WgsSWgVDo3xaVYOWLBYhs4UrvXrnDv0XDlfU7q7cmB5cQUIPnysMUN/Jtl/XokUY0ENUr6hgMHuYQc1FRV2l61OYtAuk61oc4pWNqAjBsVGoElGW+BqUITIz1nHGrV" } }, { "type": "step", "result": "=24\\left(0+0+0+376\\right)" }, { "type": "step", "primary": "Add the numbers: $$0+0+0+376=376$$", "result": "=24\\cdot\\:376" }, { "type": "step", "primary": "Multiply the numbers: $$24\\cdot\\:376=9024$$", "result": "=9024" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79oQEVzHgN4zdIGLL9QVK19jZEhTh89jESLuhDb1AwV4Wfyx43Vy6LjPXG3u/POCsx3z9op9yflolLdphqqxre6z+Dtk8aiVipEtXOuB78/dbCm9JtPvow6i0bNVwBqxuDFAy4dP0pROrQc9m7QA8BN6GQqufR6tr2vPxOUv7H++2KeLpi+mJm2bDp519ayDMp1Z6nhTpTY7AoOmanWYM0h0uLNMZSz6iaqPprG/muSYf9tYmcdjas7kEn4dJTfYzAknCqmGPKXE9Fub8CH/33uKsJPyd9/AXu9kt6jO4+lp7NRpOG5hWjzv5xG+9DMHyvZXOVtMOyFTc/mzO2Ff+lGAVmiIFo0pYVBGxNR6ZZlMkt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "=-1088+9024" }, { "type": "step", "primary": "Simplify" }, { "type": "step", "result": "=7936" } ], "meta": { "solvingClass": "Solver", "interimType": "Laplace Eval Function At Point Specific 3Eq" } }, { "type": "step", "result": "7936" } ], "meta": { "interimType": "N/A" } } ], "meta": { "interimType": "Taylor Evaluate Derivatives 0Eq" } }, { "type": "step", "result": "=0+\\frac{1}{1!}θ+\\frac{0}{2!}θ^{2}+\\frac{2}{3!}θ^{3}+\\frac{0}{4!}θ^{4}+\\frac{16}{5!}θ^{5}+\\frac{0}{6!}θ^{6}+\\frac{272}{7!}θ^{7}+\\frac{0}{8!}θ^{8}+\\frac{7936}{9!}θ^{9}+\\ldots\\:" }, { "type": "step", "primary": "Refine", "result": "=θ+\\frac{1}{3}θ^{3}+\\frac{2}{15}θ^{5}+\\frac{17}{315}θ^{7}+\\frac{62}{2835}θ^{9}+\\ldots\\:" } ], "meta": { "solvingClass": "Taylor" } }, "meta": { "showVerify": true } }