tangent of f(x)=x(27x^{-2}+2),\at x=-3

{ "query": { "display": "tangent of $$f\\left(x\\right)=x\\left(27x^{-2}+2\\right),\\:\\at\\:x=-3$$", "symbolab_question": "PRE_CALC#tangent f(x)=x(27x^{-2}+2),\\at x=-3" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Derivative Applications", "subTopic": "Tangent", "default": "y=-x-18", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Tangent line to $$f\\left(x\\right)=x\\left(27x^{-2}+2\\right)$$, at $$x=-3:{\\quad}y=-x-18$$", "steps": [ { "type": "interim", "title": "Find the tangent point:$${\\quad}\\left(-3,\\:-15\\right)$$", "steps": [ { "type": "step", "primary": "Plug $$x=-3$$ into the equation $$f\\left(x\\right)=x\\left(27x^{-2}+2\\right)$$", "result": "f\\left(x\\right)=\\left(-3\\right)\\left(27\\left(-3\\right)^{-2}+2\\right)" }, { "type": "step", "primary": "Solve $$f\\left(x\\right)$$", "result": "f\\left(x\\right)=-15" } ], "meta": { "interimType": "Tangent Find Tangent Point Title 0Eq" } }, { "type": "interim", "title": "Find the slope of $$f\\left(x\\right)=x\\left(27x^{-2}+2\\right):{\\quad}\\frac{df\\left(x\\right)}{dx}=-\\frac{27}{x^{2}}+2$$", "input": "f\\left(x\\right)=x\\left(27x^{-2}+2\\right)", "steps": [ { "type": "step", "primary": "In order to find the slope of the function, take the derivative of $$x\\left(27x^{-2}+2\\right)$$" }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(x\\left(27x^{-2}+2\\right)\\right)=-\\frac{27}{x^{2}}+2$$", "input": "\\frac{d}{dx}\\left(x\\left(27x^{-2}+2\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the Product Rule: $$\\left(f{\\cdot}g\\right)'=f'{\\cdot}g+f{\\cdot}g'$$", "secondary": [ "$$f=x,\\:g=27x^{-2}+2$$" ], "result": "=\\frac{dx}{dx}\\left(27x^{-2}+2\\right)+\\frac{d}{dx}\\left(27x^{-2}+2\\right)x", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Product%20Rule", "practiceTopic": "Product Rule" } }, { "type": "interim", "title": "$$\\frac{dx}{dx}=1$$", "input": "\\frac{dx}{dx}", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{dx}{dx}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYko/29fz701XcRtz4b42RqRjqLYrB3CcI0Y7zGHBJCja+8ZDu8iF4MSewt4yms1lIdz2XHFZ6BxfaHSMA6lT+lbVmoiKRd+ttkZ9NIrGodT+" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(27x^{-2}+2\\right)=-\\frac{54}{x^{3}}$$", "input": "\\frac{d}{dx}\\left(27x^{-2}+2\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{d}{dx}\\left(27x^{-2}\\right)+\\frac{d}{dx}\\left(2\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(27x^{-2}\\right)=-\\frac{54}{x^{3}}$$", "input": "\\frac{d}{dx}\\left(27x^{-2}\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=27\\frac{d}{dx}\\left(x^{-2}\\right)" }, { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=27\\left(-2x^{-2-1}\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "interim", "title": "Simplify $$27\\left(-2x^{-2-1}\\right):{\\quad}-\\frac{54}{x^{3}}$$", "input": "27\\left(-2x^{-2-1}\\right)", "result": "=-\\frac{54}{x^{3}}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=-27\\cdot\\:2x^{-2-1}" }, { "type": "step", "primary": "Subtract the numbers: $$-2-1=-3$$", "result": "=-27\\cdot\\:2x^{-3}" }, { "type": "step", "primary": "Apply exponent rule: $$a^{-b}=\\frac{1}{a^b}$$", "secondary": [ "$$x^{-3}=\\frac{1}{x^{3}}$$" ], "result": "=27\\cdot\\:2\\cdot\\:\\frac{1}{x^{3}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=-\\frac{1\\cdot\\:27\\cdot\\:2}{x^{3}}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:27\\cdot\\:2=54$$", "result": "=-\\frac{54}{x^{3}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7+e2TKvyMqdX30URq4hHToACWKUbvV6WK3fDUgFtg3Q+Vu0y/nymFFvapFvN0D5+8VpAF7wuGh86ExWG/nJwmC2RLd2VwIqlBNByF6663syR2SpdpleAJc7YgKUwBYoM993SS9hKl0/lDeOxOve1Te7CI2sSeA74029n2yo277ZU=" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(2\\right)=0$$", "input": "\\frac{d}{dx}\\left(2\\right)", "steps": [ { "type": "step", "primary": "Derivative of a constant: $$\\frac{d}{dx}\\left({a}\\right)=0$$", "result": "=0" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYiiraNd5UTAiEFXslV0UVyVJ8Vk6wvKjVnTtwWT18bQnz7FeFrf3rcM8IZlDz2c0dm5O2bEw0Ql6ne7k1AUriTtRm0l+ci6m9OnlYfI6EjHe" } }, { "type": "step", "result": "=-\\frac{54}{x^{3}}+0" }, { "type": "step", "primary": "Simplify", "result": "=-\\frac{54}{x^{3}}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=1\\cdot\\:\\left(27x^{-2}+2\\right)+\\left(-\\frac{54}{x^{3}}\\right)x" }, { "type": "interim", "title": "Simplify $$1\\cdot\\:\\left(27x^{-2}+2\\right)+\\left(-\\frac{54}{x^{3}}\\right)x:{\\quad}-\\frac{27}{x^{2}}+2$$", "input": "1\\cdot\\:\\left(27x^{-2}+2\\right)+\\left(-\\frac{54}{x^{3}}\\right)x", "result": "=-\\frac{27}{x^{2}}+2", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=1\\cdot\\:\\left(27x^{-2}+2\\right)-\\frac{54}{x^{3}}x" }, { "type": "interim", "title": "$$1\\cdot\\:\\left(27x^{-2}+2\\right)=\\frac{27}{x^{2}}+2$$", "input": "1\\cdot\\:\\left(27x^{-2}+2\\right)", "steps": [ { "type": "interim", "title": "$$27x^{-2}=\\frac{27}{x^{2}}$$", "input": "27x^{-2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{-b}=\\frac{1}{a^b}$$", "secondary": [ "$$x^{-2}=\\frac{1}{x^{2}}$$" ], "result": "=27\\cdot\\:\\frac{1}{x^{2}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:27}{x^{2}}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:27=27$$", "result": "=\\frac{27}{x^{2}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7rWX1zZIw2+o5VQx1j+vgms0ag8T1MwTer44+aCS/ZFDOkL2++fbIeJ/F/C21a0+Fe03RoCbBonT9nPSN8LY9xANkysly19C71EEkNzIp7IZxbigBv5uYjY0te7HRxG0+FkVDw5yEF2DEw4msQzKTew==" } }, { "type": "step", "result": "=1\\cdot\\:\\left(\\frac{27}{x^{2}}+2\\right)" }, { "type": "step", "primary": "Refine", "result": "=\\frac{27}{x^{2}}+2" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s73dGQ1SEbLzg0wnb3kzwQwJJMeo4ZA2D3YVvVielviyyrju+5Z51e/ZZSD3gRHwjBZ2donTHEbc0OkcDIHLHDYNFax2S0u4YEweKenWTuby8PN+L9byZbCx6C/ZCko45/fwvFYd5mCWz8lH1q/zBamQoDu3JE0UG9ftV5ON3tFTkkt3WiGR7ZaCaXvz77bMjS" } }, { "type": "interim", "title": "$$\\frac{54}{x^{3}}x=\\frac{54}{x^{2}}$$", "input": "\\frac{54}{x^{3}}x", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{54x}{x^{3}}" }, { "type": "step", "primary": "Cancel the common factor: $$x$$", "result": "=\\frac{54}{x^{2}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7abCqHyALV5qVttvw0EQTuOz7exmd3mqJNOkcV84v2LzMwViaLUXkeD+JukROhWdj+hawWsKq0DVoqWKvxbk1BAH2kDe5DGYTz3TrPquGdIimLh8ybGlccIEwDrixwoNc1hmkXVORR1vWWAGdzcaYvFypnVfW1elSOrolhvXW8DI=" } }, { "type": "step", "result": "=\\frac{27}{x^{2}}+2-\\frac{54}{x^{2}}" }, { "type": "interim", "title": "Combine the fractions $$\\frac{27}{x^{2}}-\\frac{54}{x^{2}}:{\\quad}-\\frac{27}{x^{2}}$$", "result": "=-\\frac{27}{x^{2}}+2", "steps": [ { "type": "step", "primary": "Apply rule $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{27-54}{x^{2}}" }, { "type": "step", "primary": "Subtract the numbers: $$27-54=-27$$", "result": "=\\frac{-27}{x^{2}}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{27}{x^{2}}" } ], "meta": { "interimType": "LCD Top Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s73dGQ1SEbLzg0wnb3kzwQwBc4Flr7SIlIDZ1zDJ1l5TGdqn9IR3i6OupdNUkccjxCICf2WQN9mSJxQaQ7cQX4iplokxKUI2rd0ju/5dpufJfmxwILlp/N2fF/0176fLJVRSpN33oxZMojoqvYhvSJACLkpiVA7S8UT8ieezZKu6plZ2/Is1XorHzGlwM7E7kG54rPpp+Si8m0YQHjRhVZt13d+4J3P3BtuUFS/Dk1MXc=" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "-\\frac{27}{x^{2}}+2" } ], "meta": { "interimType": "Slope Equation Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMGyxztx1DIZ4Y9QoeLjXWS0V72qDfdFWDFOnT7qFJwJJTU2Y+DM/tPp5bmOVP4xs1bkoH6MHGg0mtOsg27VY89g7ggd6D8X+XQG2x+mOeg3dyZyN7U6QxHyize+A3z4dK4gBJl4WMO1rA0a30/bUYlg==" } }, { "type": "interim", "title": "$$EN:\\:Title\\:General\\:Equation\\:Slope\\:At\\:Point\\:2Eq:{\\quad}m=-1$$", "steps": [ { "type": "step", "primary": "Plug $$x=-3$$ into the equation $$-\\frac{27}{x^{2}}+2$$", "result": "-\\frac{27}{\\left(-3\\right)^{2}}+2" }, { "type": "interim", "title": "Simplify $$-\\frac{27}{\\left(-3\\right)^{2}}+2:{\\quad}-1$$", "input": "-\\frac{27}{\\left(-3\\right)^{2}}+2", "result": "=-1", "steps": [ { "type": "interim", "title": "$$\\frac{27}{\\left(-3\\right)^{2}}=3$$", "input": "\\frac{27}{\\left(-3\\right)^{2}}", "steps": [ { "type": "interim", "title": "$$\\left(-3\\right)^{2}=3^{2}$$", "input": "\\left(-3\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=a^{n},\\:$$if $$n$$ is even", "secondary": [ "$$\\left(-3\\right)^{2}=3^{2}$$" ], "result": "=3^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7DiFM8Xt0cYQ3q7JWVRPZcM0ag8T1MwTer44+aCS/ZFA8nbDyNbrSnrFcItTWaPZwZtf025QjpwjfUn/kUwetFOiKbjfIuZ59Og03MnErgqY=" } }, { "type": "step", "result": "=\\frac{27}{3^{2}}" }, { "type": "interim", "title": "Factor $$27:{\\quad}3^{3}$$", "steps": [ { "type": "step", "primary": "Factor $$27=3^{3}$$" } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "=\\frac{3^{3}}{3^{2}}" }, { "type": "interim", "title": "Cancel $$\\frac{3^{3}}{3^{2}}:{\\quad}3$$", "input": "\\frac{3^{3}}{3^{2}}", "result": "=3", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\frac{x^{a}}{x^{b}}=x^{a-b}$$", "secondary": [ "$$\\frac{3^{3}}{3^{2}}=3^{3-2}$$" ], "result": "=3^{3-2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Subtract the numbers: $$3-2=1$$", "result": "=3" } ], "meta": { "interimType": "Generic Cancel Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYhmHuw74KjGEn9LyLcld7BVV00rpv8+ZC6TM10tVCSHsWS0g+IDDIZeR6y3tPkPK7PC30sSftAIFS6Qkpy19IkrfcvtK1wS5X9djgJbBX+NYdvAtMv+mbZsemSO0yGcaKb8yD3hLQ33B7/8/LpbPE3o=" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s72fmOVsAXDoJz5uyQ4nWLkwbeC2QCdLxKEECRBZFZMUSrju+5Z51e/ZZSD3gRHwjBUP9XflYaQHHpNIh9mK9vyhKejH85OvhPZapB93H0SrkUoFXiX1jlhc7DGNGGqEDO" } }, { "type": "step", "result": "=-3+2" }, { "type": "step", "primary": "Add/Subtract the numbers: $$-3+2=-1$$", "result": "=-1" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Ph4Hdpt9b0sN3xgfermWkleBT68G3COWpGxP5sghghoDnzlbPZjyKgy1eUCFsLd5pmhpfHqzxU6UVoC7gLWdY3ql8XXPq6bNQlMm+36iNhm4ztFaEsONSrO2lT70vQjOaTKlK61ugVTIVrW6bMRlOHjp3RZ6ZZVo/20EO1XzzH0=" } }, { "type": "step", "result": "m=-1" } ], "meta": { "interimType": "General Equation Slope At Point 2Eq" } }, { "type": "interim", "title": "Find the line with slope m=$$-1$$ and passing through $$\\left(-3,\\:-15\\right):{\\quad}y=-x-18$$", "steps": [ { "type": "step", "primary": "Compute the line equation $$\\mathbf{y=mx+b}$$ for slope m=$$-1$$ and passing through $$\\left(-3,\\:-15\\right)$$" }, { "type": "interim", "title": "Compute the $$y$$ intercept:$${\\quad}b=-18$$", "steps": [ { "type": "step", "primary": "Plug the slope $$-1$$ into $$y=mx+b$$", "result": "y=\\left(-1\\right)x+b" }, { "type": "step", "primary": "Plug in $$\\left(-3,\\:-15\\right)$$: $$\\quad\\:x=-3,\\:y=-15$$", "result": "-15=\\left(-1\\right)\\left(-3\\right)+b" }, { "type": "step", "primary": "Isolate $$b$$" }, { "type": "interim", "title": "$$-15=\\left(-1\\right)\\left(-3\\right)+b{\\quad:\\quad}b=-18$$", "input": "-15=\\left(-1\\right)\\left(-3\\right)+b", "steps": [ { "type": "step", "primary": "Switch sides", "result": "\\left(-1\\right)\\left(-3\\right)+b=-15" }, { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a,\\:-\\left(-a\\right)=a$$", "result": "1\\cdot\\:3+b=-15" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:3=3$$", "result": "3+b=-15" }, { "type": "interim", "title": "Move $$3\\:$$to the right side", "input": "3+b=-15", "result": "b=-18", "steps": [ { "type": "step", "primary": "Subtract $$3$$ from both sides", "result": "3+b-3=-15-3" }, { "type": "step", "primary": "Simplify", "result": "b=-18" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "result": "b=-18" } ], "meta": { "interimType": "Line Equation Find Intersection From Point 0Eq" } }, { "type": "step", "primary": "Construct the line equation $$\\mathbf{y=mx+b}$$ where $$\\mathbf{m}=-1$$ and $$\\mathbf{b}=-18$$", "result": "y=-x-18" } ], "meta": { "interimType": "Line Equation Slope Point 6Eq" } }, { "type": "step", "result": "y=-x-18" } ], "meta": { "solvingClass": "PreCalc" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "tangent f(x)=x(27x^{-2}+2),\\at x=-3" }, "showViewLarger": true } }, "meta": { "showVerify": true } }