tangent of y=4(x-1/x)^3,\at x=2

{ "query": { "display": "tangent of $$y=4\\left(x-\\frac{1}{x}\\right)^{3},\\:\\at\\:x=2$$", "symbolab_question": "PRE_CALC#tangent y=4(x-\\frac{1}{x})^{3},\\at x=2" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Derivative Applications", "subTopic": "Tangent", "default": "y=\\frac{135}{4}x-54", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Tangent line to $$y=4\\left(x-\\frac{1}{x}\\right)^{3}$$, at $$x=2:{\\quad}y=\\frac{135}{4}x-54$$", "steps": [ { "type": "interim", "title": "Find the tangent point:$${\\quad}\\left(2,\\:\\frac{27}{2}\\right)$$", "steps": [ { "type": "step", "primary": "Plug $$x=2$$ into the equation $$y=4\\left(x-\\frac{1}{x}\\right)^{3}$$", "result": "y=4\\left(2-\\frac{1}{2}\\right)^{3}" }, { "type": "step", "primary": "Solve $$y$$", "result": "y=\\frac{27}{2}" } ], "meta": { "interimType": "Tangent Find Tangent Point Title 0Eq" } }, { "type": "interim", "title": "Find the slope of $$y=4\\left(x-\\frac{1}{x}\\right)^{3}:{\\quad}\\frac{dy}{dx}=12x^{2}+\\frac{12}{x^{4}}-\\frac{12}{x^{2}}-12$$", "input": "y=4\\left(x-\\frac{1}{x}\\right)^{3}", "steps": [ { "type": "step", "primary": "In order to find the slope of the function, take the derivative of $$4\\left(x-\\frac{1}{x}\\right)^{3}$$" }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(4\\left(x-\\frac{1}{x}\\right)^{3}\\right)=12x^{2}+\\frac{12}{x^{4}}-\\frac{12}{x^{2}}-12$$", "input": "\\frac{d}{dx}\\left(4\\left(x-\\frac{1}{x}\\right)^{3}\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=4\\frac{d}{dx}\\left(\\left(x-\\frac{1}{x}\\right)^{3}\\right)" }, { "type": "interim", "title": "Apply the chain rule:$${\\quad}3\\left(x-\\frac{1}{x}\\right)^{2}\\frac{d}{dx}\\left(x-\\frac{1}{x}\\right)$$", "input": "\\frac{d}{dx}\\left(\\left(x-\\frac{1}{x}\\right)^{3}\\right)", "result": "=3\\left(x-\\frac{1}{x}\\right)^{2}\\frac{d}{dx}\\left(x-\\frac{1}{x}\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{3},\\:\\:u=\\left(x-\\frac{1}{x}\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{3}\\right)\\frac{d}{dx}\\left(\\left(x-\\frac{1}{x}\\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}{dx}\\left(\\left(x-\\frac{1}{x}\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\left(x-\\frac{1}{x}\\right)$$", "result": "=3\\left(x-\\frac{1}{x}\\right)^{2}\\frac{d}{dx}\\left(x-\\frac{1}{x}\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYhrb549AcNeKw47mEBzrnqL3eisivVfnk/N487uIc+aESUM9pakkKILvT6Fs/PM352UPWd0or+4i3R6mQ374eeZpkpAtzMPRA2gXdSSjY8vEQllyn/Q+3qZWsmBdfv3za/mAxv96za1jtB37K2LJXI9XhBp+uCH0/T09rsvJgzQWKjTttt5tJ7d0pMqV6FW50AS4M5VpC8qh+oehjmM1qmzPHVJGaR3CuIp5NX3rLDDQialcV/dI5TH4fXyp+ncwuA==" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(x-\\frac{1}{x}\\right)=1+\\frac{1}{x^{2}}$$", "input": "\\frac{d}{dx}\\left(x-\\frac{1}{x}\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{dx}{dx}-\\frac{d}{dx}\\left(\\frac{1}{x}\\right)" }, { "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": "$$\\frac{d}{dx}\\left(\\frac{1}{x}\\right)=-\\frac{1}{x^{2}}$$", "input": "\\frac{d}{dx}\\left(\\frac{1}{x}\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\frac{1}{a}=a^{-1}$$", "result": "=\\frac{d}{dx}\\left(x^{-1}\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=-1\\cdot\\:x^{-1-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "interim", "title": "Simplify $$-1\\cdot\\:x^{-1-1}:{\\quad}-\\frac{1}{x^{2}}$$", "input": "-1\\cdot\\:x^{-1-1}", "result": "=-\\frac{1}{x^{2}}", "steps": [ { "type": "step", "primary": "Subtract the numbers: $$-1-1=-2$$", "result": "=-1\\cdot\\:x^{-2}" }, { "type": "step", "primary": "Apply exponent rule: $$a^{-b}=\\frac{1}{a^b}$$", "secondary": [ "$$x^{-2}=\\frac{1}{x^{2}}$$" ], "result": "=-1\\cdot\\:\\frac{1}{x^{2}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:\\frac{1}{x^{2}}=\\frac{1}{x^{2}}$$", "result": "=-\\frac{1}{x^{2}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7+AgNZzeIicTfbr51/JGq1KiJTJQkRRngZl07rbqjeC6jkVi15I8rBefLi4Iyt2wrT1r2iaXj0z6hiHFOta70Gf8//6/nV5O4fb8Xgwi7mapyhd7tjiG+GxQNxDvGkZUl/dfOQfCkILEECUyPIy9DrRF1+E4wvPRIGnJs5KwUnrw=" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=1-\\left(-\\frac{1}{x^{2}}\\right)" }, { "type": "step", "primary": "Simplify", "result": "=1+\\frac{1}{x^{2}}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=4\\cdot\\:3\\left(x-\\frac{1}{x}\\right)^{2}\\left(1+\\frac{1}{x^{2}}\\right)" }, { "type": "interim", "title": "Simplify $$4\\cdot\\:3\\left(x-\\frac{1}{x}\\right)^{2}\\left(1+\\frac{1}{x^{2}}\\right):{\\quad}12x^{2}+\\frac{12}{x^{4}}-\\frac{12}{x^{2}}-12$$", "input": "4\\cdot\\:3\\left(x-\\frac{1}{x}\\right)^{2}\\left(1+\\frac{1}{x^{2}}\\right)", "result": "=12x^{2}+\\frac{12}{x^{4}}-\\frac{12}{x^{2}}-12", "steps": [ { "type": "interim", "title": "$$\\left(x-\\frac{1}{x}\\right)^{2}=x^{2}-2+\\frac{1}{x^{2}}$$", "input": "\\left(x-\\frac{1}{x}\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply Perfect Square Formula: $$\\left(a-b\\right)^{2}=a^{2}-2ab+b^{2}$$", "secondary": [ "$$a=x,\\:\\:b=\\frac{1}{x}$$" ], "meta": { "practiceLink": "/practice/expansion-practice#area=main&subtopic=Perfect%20Square", "practiceTopic": "Expand Perfect Square" } }, { "type": "step", "result": "=x^{2}-2x\\frac{1}{x}+\\left(\\frac{1}{x}\\right)^{2}" }, { "type": "interim", "title": "Simplify $$x^{2}-2x\\frac{1}{x}+\\left(\\frac{1}{x}\\right)^{2}:{\\quad}x^{2}-2+\\frac{1}{x^{2}}$$", "input": "x^{2}-2x\\frac{1}{x}+\\left(\\frac{1}{x}\\right)^{2}", "result": "=x^{2}-2+\\frac{1}{x^{2}}", "steps": [ { "type": "interim", "title": "$$2x\\frac{1}{x}=2$$", "input": "2x\\frac{1}{x}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:2x}{x}" }, { "type": "step", "primary": "Cancel the common factor: $$x$$", "result": "=1\\cdot\\:2" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:2=2$$", "result": "=2" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7YWc8v2J/bdwhe+aLTy5S+ACWKUbvV6WK3fDUgFtg3Q8kLk5B3NY35r4MgzhzVzFVdo32iBGpFtgPE3KhTebUJOKJwc5B82N2ip9PG0B4bWYkt3WiGR7ZaCaXvz77bMjS" } }, { "type": "interim", "title": "$$\\left(\\frac{1}{x}\\right)^{2}=\\frac{1}{x^{2}}$$", "input": "\\left(\\frac{1}{x}\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(\\frac{a}{b}\\right)^{c}=\\frac{a^{c}}{b^{c}}$$", "result": "=\\frac{1^{2}}{x^{2}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "secondary": [ "$$1^{2}=1$$" ], "result": "=\\frac{1}{x^{2}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s78f5zsOjIL/iTUCSjOJw3YY5IpdliG1E4K4EtDGLN9yvMwViaLUXkeD+JukROhWdji/4oeYZyQACKxTaxbZBy5P8//6/nV5O4fb8Xgwi7maqowaES/eGNGT791M9pqGZciPdRac0QdcAT+tD2Uj5Yk1iVI3uvN1by+AN9NfjoKFU=" } }, { "type": "step", "result": "=x^{2}-2+\\frac{1}{x^{2}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/Si+JDA8R6NsKW8o4twqQUBgnzAZkUy+vMtgHHTjQTCrju+5Z51e/ZZSD3gRHwjBD/qS5oU9anC+30lxPs+LPL3EC6fzpSTQVL8rGcc/O8uGp7i1v9jIOeuGn+FcrnmkgSMBgdgp2UYObPABvJV5EOVwTjmTNVzpA/S5/ME/ZN/NgDPG02alGZtmlvpfqCA7" } }, { "type": "step", "result": "=4\\cdot\\:3\\left(x^{2}+\\frac{1}{x^{2}}-2\\right)\\left(\\frac{1}{x^{2}}+1\\right)" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:3=12$$", "result": "=12\\left(x^{2}+\\frac{1}{x^{2}}-2\\right)\\left(\\frac{1}{x^{2}}+1\\right)" }, { "type": "interim", "title": "Expand $$\\left(x^{2}-2+\\frac{1}{x^{2}}\\right)\\left(1+\\frac{1}{x^{2}}\\right):{\\quad}x^{2}+\\frac{1}{x^{4}}-\\frac{1}{x^{2}}-1$$", "input": "\\left(x^{2}-2+\\frac{1}{x^{2}}\\right)\\left(1+\\frac{1}{x^{2}}\\right)", "result": "=12\\left(x^{2}+\\frac{1}{x^{4}}-\\frac{1}{x^{2}}-1\\right)", "steps": [ { "type": "step", "primary": "Distribute parentheses", "result": "=x^{2}\\cdot\\:1+x^{2}\\frac{1}{x^{2}}+\\left(-2\\right)\\cdot\\:1+\\left(-2\\right)\\frac{1}{x^{2}}+\\frac{1}{x^{2}}\\cdot\\:1+\\frac{1}{x^{2}}\\cdot\\:\\frac{1}{x^{2}}", "meta": { "title": { "extension": "Multiply each of the terms within the first parentheses<br/>by each of the terms within the second parentheses left to right" } } }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$+\\left(-a\\right)=-a$$" ], "result": "=1\\cdot\\:x^{2}+\\frac{1}{x^{2}}x^{2}-2\\cdot\\:1-2\\cdot\\:\\frac{1}{x^{2}}+1\\cdot\\:\\frac{1}{x^{2}}+\\frac{1}{x^{2}}\\cdot\\:\\frac{1}{x^{2}}" }, { "type": "interim", "title": "Simplify $$1\\cdot\\:x^{2}+\\frac{1}{x^{2}}x^{2}-2\\cdot\\:1-2\\cdot\\:\\frac{1}{x^{2}}+1\\cdot\\:\\frac{1}{x^{2}}+\\frac{1}{x^{2}}\\cdot\\:\\frac{1}{x^{2}}:{\\quad}x^{2}+\\frac{1}{x^{4}}-\\frac{1}{x^{2}}-1$$", "input": "1\\cdot\\:x^{2}+\\frac{1}{x^{2}}x^{2}-2\\cdot\\:1-2\\cdot\\:\\frac{1}{x^{2}}+1\\cdot\\:\\frac{1}{x^{2}}+\\frac{1}{x^{2}}\\cdot\\:\\frac{1}{x^{2}}", "result": "=x^{2}+\\frac{1}{x^{4}}-\\frac{1}{x^{2}}-1", "steps": [ { "type": "step", "primary": "Add similar elements: $$-2\\cdot\\:\\frac{1}{x^{2}}+1\\cdot\\:\\frac{1}{x^{2}}=-\\frac{1}{x^{2}}$$", "result": "=1\\cdot\\:x^{2}+1\\cdot\\:\\frac{1}{x^{2}}x^{2}-2\\cdot\\:1-\\frac{1}{x^{2}}+1\\cdot\\:\\frac{1}{x^{2}}\\cdot\\:\\frac{1}{x^{2}}" }, { "type": "interim", "title": "$$1\\cdot\\:x^{2}=x^{2}$$", "input": "1\\cdot\\:x^{2}", "steps": [ { "type": "step", "primary": "Multiply: $$1\\cdot\\:x^{2}=x^{2}$$", "result": "=x^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s78hVycPB+rfQCHnyh7m3HTi061ljBSPJeENOw2efoSWt5uKsWh4cBdpi/wldLLf2V/z//r+dXk7h9vxeDCLuZqsWPXBkUr6zzMsua0zkIRDrF3DenEWojLSGXYMDcAl7b" } }, { "type": "interim", "title": "$$1\\cdot\\:\\frac{1}{x^{2}}x^{2}=1$$", "input": "1\\cdot\\:\\frac{1}{x^{2}}x^{2}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=1\\cdot\\:\\frac{1\\cdot\\:x^{2}}{x^{2}}" }, { "type": "step", "primary": "Cancel the common factor: $$x^{2}$$", "result": "=1\\cdot\\:1" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:1=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s77FHIuRV2uKBi3mFgtfT9VJMy13JI+809gskqph1Hm4ItOtZYwUjyXhDTsNnn6ElrPGrllHeFjaKLCz4xpIb40aBzIB6qdky9r1WITjaDuDTFDfyC/Jzv7uiABZ+ezuwy8DKIcIyGlDqoMci7lrAOjA==" } }, { "type": "interim", "title": "$$2\\cdot\\:1=2$$", "input": "2\\cdot\\:1", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=2" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7+PAYxXmLR4nDEBHt2jGqsN13jtrSFDx+UNsawjlOjV3jAewWnbvHwHHNJ9dhy4+WsiFkFLY2UCqxUg31WjmL6HgMxnm5w+rT3GGDPdan9jw=" } }, { "type": "interim", "title": "$$1\\cdot\\:\\frac{1}{x^{2}}\\cdot\\:\\frac{1}{x^{2}}=\\frac{1}{x^{4}}$$", "input": "1\\cdot\\:\\frac{1}{x^{2}}\\cdot\\:\\frac{1}{x^{2}}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$\\frac{a}{b}\\cdot\\frac{c}{d}=\\frac{a\\:\\cdot\\:c}{b\\:\\cdot\\:d}$$", "result": "=1\\cdot\\:\\frac{1\\cdot\\:1}{x^{2}x^{2}}" }, { "type": "interim", "title": "Simplify $$\\frac{1\\cdot\\:1}{x^{2}x^{2}}:{\\quad}\\frac{1}{x^{4}}$$", "input": "\\frac{1\\cdot\\:1}{x^{2}x^{2}}", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:1=1$$", "result": "=\\frac{1}{x^{2}x^{2}}" }, { "type": "interim", "title": "$$x^{2}x^{2}=x^{4}$$", "input": "x^{2}x^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$x^{2}x^{2}=\\:x^{2+2}$$" ], "result": "=x^{2+2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+2=4$$", "result": "=x^{4}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s708rgV9eva25+03lnkIPLfCAn9lkDfZkicUGkO3EF+IodP1z01S2yZ8KErqMvBOgMyGKp9VhzCo0zZo4xoLjx0L4bX+TSoLgj4qmKpRCz5kQkt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "=\\frac{1}{x^{4}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=1\\cdot\\:\\frac{1}{x^{4}}" }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:\\frac{1}{x^{4}}=\\frac{1}{x^{4}}$$", "result": "=\\frac{1}{x^{4}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s77FHIuRV2uKBi3mFgtfT9VDaTG1mRx8XZrqe5uXHFCb+d2m4NDAtb8VqK336+NToTfJEe4bDv60zkwRNMOPq9UULJUwysTkf+ifmThr1HGroSun1uzfO2K3AUYRqF8CakxRWVzZt0HhmQuzJong/s7SQk2nuCPF/oGhhWM0M4a9rfABEmfnTYCOdLSnnDGgSiS6vAzagImrjWZ520oY0fmCcN0nMyjtSCncdtXyti8Sokt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "=x^{2}+1-2-\\frac{1}{x^{2}}+\\frac{1}{x^{4}}" }, { "type": "step", "primary": "Add/Subtract the numbers: $$1-2=-1$$", "result": "=x^{2}+\\frac{1}{x^{4}}-\\frac{1}{x^{2}}-1" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7+39AaWf//1+EMA42i6i9igu2j1feoxvYUT3h6cDBbGuQLYS1773XNNkQo4YqtLqnAJYpRu9XpYrd8NSAW2DdD3k4sBLE/lfaqPAZYFcr8Xz5sJl5WdmnHDUFZpwekO4khaZIR2npB1kSZbKIMJD0OMJ03PG0mVy6c9NMH4lWvrVAy41+w3GLQeeKt4HkTXoxiBM2TU2wja4O+bQ98HtYs7TLhe8t2tsverUFqsrJqp8S9E/XlQOcNqB0EHLcGgLdc3vTmoOSUSs6vELS8zFdSg==" } }, { "type": "interim", "title": "Expand $$12\\left(x^{2}+\\frac{1}{x^{4}}-\\frac{1}{x^{2}}-1\\right):{\\quad}12x^{2}+\\frac{12}{x^{4}}-\\frac{12}{x^{2}}-12$$", "input": "12\\left(x^{2}+\\frac{1}{x^{4}}-\\frac{1}{x^{2}}-1\\right)", "result": "=12x^{2}+\\frac{12}{x^{4}}-\\frac{12}{x^{2}}-12", "steps": [ { "type": "step", "primary": "Distribute parentheses", "result": "=12x^{2}+12\\cdot\\:\\frac{1}{x^{4}}+12\\left(-\\frac{1}{x^{2}}\\right)+12\\left(-1\\right)", "meta": { "title": { "extension": "Multiply each of the terms within the parentheses<br/>by the term outside the parenthesis" } } }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$+\\left(-a\\right)=-a$$" ], "result": "=12x^{2}+12\\cdot\\:\\frac{1}{x^{4}}-12\\cdot\\:\\frac{1}{x^{2}}-12\\cdot\\:1" }, { "type": "interim", "title": "Simplify $$12x^{2}+12\\cdot\\:\\frac{1}{x^{4}}-12\\cdot\\:\\frac{1}{x^{2}}-12\\cdot\\:1:{\\quad}12x^{2}+\\frac{12}{x^{4}}-\\frac{12}{x^{2}}-12$$", "input": "12x^{2}+12\\cdot\\:\\frac{1}{x^{4}}-12\\cdot\\:\\frac{1}{x^{2}}-12\\cdot\\:1", "result": "=12x^{2}+\\frac{12}{x^{4}}-\\frac{12}{x^{2}}-12", "steps": [ { "type": "interim", "title": "$$12\\cdot\\:\\frac{1}{x^{4}}=\\frac{12}{x^{4}}$$", "input": "12\\cdot\\:\\frac{1}{x^{4}}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:12}{x^{4}}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:12=12$$", "result": "=\\frac{12}{x^{4}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7X9Re+PckSoemCIRQQUfDdc5VHEIrypybN3rzU3DsWNjdd47a0hQ8flDbGsI5To1doDDfDl9rb93jWiqTfsVP2ld5Jw/F3nvI+3MovIv55IdX2fJJMHaN/8mrcV3OwjRRoJBDm9+s0zNjOHUWqICkPDawn8xq2c0tK1eQxI5qrmy4azBCZOGbaHNTuYhYOa9+" } }, { "type": "interim", "title": "$$12\\cdot\\:\\frac{1}{x^{2}}=\\frac{12}{x^{2}}$$", "input": "12\\cdot\\:\\frac{1}{x^{2}}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:12}{x^{2}}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:12=12$$", "result": "=\\frac{12}{x^{2}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7X9Re+PckSoemCIRQQUfDdfvjpzxlnNUDEWutROrcGY7dd47a0hQ8flDbGsI5To1doDDfDl9rb93jWiqTfsVP2m82tbTdGuGAq/Y6SAYhNw9X2fJJMHaN/8mrcV3OwjRRYwS4imLhjfgZen9TNaCCojawn8xq2c0tK1eQxI5qrmxYlSN7rzdW8vgDfTX46ChV" } }, { "type": "interim", "title": "$$12\\cdot\\:1=12$$", "input": "12\\cdot\\:1", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$12\\cdot\\:1=12$$", "result": "=12" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s76snTl8vomBaze8oXS1/i0iAn9lkDfZkicUGkO3EF+Ir3gfwKgaetL3sTzsyK0NpAX3/HeVcWy6aj8lZWVvtN6eDNSO/68lw5VtrKvdHbeGk=" } }, { "type": "step", "result": "=12x^{2}+\\frac{12}{x^{4}}-\\frac{12}{x^{2}}-12" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7+5rLgi+UyuIZjMKAdU+Z/EolkgeStU8KzVqp+Dpn3fJotlzWSS+67aPDtF93einbLTrWWMFI8l4Q07DZ5+hJa21UA5jZ7zMwYSpxTlM1E+elq5PCxfP65WlHRznG4qEcwdEOS81txGEFH84iuiGCrWNrEKY+bD7kpz8jqkZtAeLwt9LEn7QCBUukJKctfSJKpVsjlHcfnEJZ5+e9Xi6c5OH/YLgW6Q33B/kMB+BRlqjh3c1c+qf2v72by8jVfB0I4y0b1cBc56bnVHqjcwaWcQ==" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7w5sJV78UfzASvVSOLTfpuf1s6oJbKXW0M5JupeKu11V8EkM1HC5iBmEcKGvd1KDK4yRCewxsmcZ0nFItBCwA46ORWLXkjysF58uLgjK3bCtXcDIUw22Y1zHgmGeDweGngkyc7EF7l3PuDUrhzSswgCUNzcSJo6TRQ3Z9/QT7bJKjeh7+jKEzLb7VNCEMF3Z/bMzoTd+5nEXVeQoBhpFcIEuaxK8PNcUGC265X/YB2zNynQu17seNZdGGiFqWst5OBFeWRnVs2eK9+UyKeibPEr++BkIi2qZNiO8mRKzoqi8=" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "12x^{2}+\\frac{12}{x^{4}}-\\frac{12}{x^{2}}-12" } ], "meta": { "interimType": "Slope Equation Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMGyxztx1DIZ4Y9QoeLjXWS76TP1+hRwf2+abqbAjotDu9SI4tLmYUsWm963Kg3K5heqXxdc+rps1CUyb7fqI2GTY2/59btFOll/rb+PYi2SlJTosUfwRfBza8a3byrOCSppS7mSekK5+AfITdTm3YWA==" } }, { "type": "interim", "title": "$$EN:\\:Title\\:General\\:Equation\\:Slope\\:At\\:Point\\:2Eq:{\\quad}m=\\frac{135}{4}$$", "steps": [ { "type": "step", "primary": "Plug $$x=2$$ into the equation $$12x^{2}+\\frac{12}{x^{4}}-\\frac{12}{x^{2}}-12$$", "result": "12\\cdot\\:2^{2}+\\frac{12}{2^{4}}-\\frac{12}{2^{2}}-12" }, { "type": "interim", "title": "Simplify $$12\\cdot\\:2^{2}+\\frac{12}{2^{4}}-\\frac{12}{2^{2}}-12:{\\quad}\\frac{135}{4}$$", "input": "12\\cdot\\:2^{2}+\\frac{12}{2^{4}}-\\frac{12}{2^{2}}-12", "result": "=\\frac{135}{4}", "steps": [ { "type": "interim", "title": "Cancel $$\\frac{12}{2^{4}}:{\\quad}\\frac{3}{2^{2}}$$", "input": "\\frac{12}{2^{4}}", "steps": [ { "type": "interim", "title": "Factor $$12:{\\quad}2^{2}\\cdot\\:3$$", "steps": [ { "type": "step", "primary": "Factor $$12=2^{2}\\cdot\\:3$$" } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "=\\frac{2^{2}\\cdot\\:3}{2^{4}}" }, { "type": "interim", "title": "Cancel $$\\frac{2^{2}\\cdot\\:3}{2^{4}}:{\\quad}\\frac{3}{2^{2}}$$", "input": "\\frac{2^{2}\\cdot\\:3}{2^{4}}", "result": "=\\frac{3}{2^{2}}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\frac{x^{a}}{x^{b}}=\\frac{1}{x^{b-a}}$$", "secondary": [ "$$\\frac{2^{2}}{2^{4}}=\\frac{1}{2^{4-2}}$$" ], "result": "=\\frac{3}{2^{4-2}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Subtract the numbers: $$4-2=2$$", "result": "=\\frac{3}{2^{2}}" } ], "meta": { "interimType": "Generic Cancel Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYu8x6oDjkSAaRFZLpQhh4lDLUvx9rKY4c2XG4pjJ5SJJq47vuWedXv2WUg94ER8IwRJaoa53T+3jKr8ayUW7wAi7e26vi5heGe/fJp9Z/1i+aVDrVFxTljmLik8W1DBFW8e6UYdRf98+HKJqvMgIPdsCqHXKE/5kGaizy/qewa8/JLd1ohke2Wgml78++2zI0g==" } } ], "meta": { "interimType": "Generic Cancel Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYl04nuwUjKDk57jPD/YGMeTdd47a0hQ8flDbGsI5To1dK6xRNRPeYgVihvos0bqhQLEkkVYT4Hty2jKgkaqKeEWc0UAm97e3ZdFs70jM7AY++kJtqarv4lgzHVhVVx8eVqwKHfe4IydqM0O9bglaye8=" } }, { "type": "step", "result": "=2^{2}\\cdot\\:12+\\frac{3}{2^{2}}-\\frac{12}{2^{2}}-12" }, { "type": "interim", "title": "Cancel $$\\frac{12}{2^{2}}:{\\quad}3$$", "input": "\\frac{12}{2^{2}}", "steps": [ { "type": "interim", "title": "Factor $$12:{\\quad}2^{2}\\cdot\\:3$$", "steps": [ { "type": "step", "primary": "Factor $$12=2^{2}\\cdot\\:3$$" } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "=\\frac{2^{2}\\cdot\\:3}{2^{2}}" }, { "type": "step", "primary": "Cancel the common factor: $$2^{2}$$", "result": "=3" } ], "meta": { "interimType": "Generic Cancel Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYtje/zIHwcpRshCql6jUKvrdd47a0hQ8flDbGsI5To1dNNwes1pEn8jD21yE6H99Wk3kCh3oevUunZ7/b0qFKBQj8UxsvvAFYSNtRTEGfD7fDq1hkJmPhVUWhTSHB7vsvg==" } }, { "type": "step", "result": "=2^{2}\\cdot\\:12+\\frac{3}{2^{2}}-3-12" }, { "type": "step", "primary": "Subtract the numbers: $$-3-12=-15$$", "result": "=\\frac{3}{2^{2}}+2^{2}\\cdot\\:12-15" }, { "type": "interim", "title": "$$\\frac{3}{2^{2}}=\\frac{3}{4}$$", "input": "\\frac{3}{2^{2}}", "steps": [ { "type": "step", "primary": "$$2^{2}=4$$", "result": "=\\frac{3}{4}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s72h2vPJACrbvlXz/1Y1bjxUV2IovRuj7SUtFTM1ebUgT9ovYKijQYhJDCbxu/nAOJoLiMg+4bsI1oqeIp4XMtHQ4bfwiV6iMLJ5sC1nL7dOaJFjD1fVI6DV4N69VhXAEmsYm0iR0P24MA8pupcVtXSQ==" } }, { "type": "interim", "title": "$$2^{2}\\cdot\\:12=48$$", "input": "2^{2}\\cdot\\:12", "steps": [ { "type": "step", "primary": "$$2^{2}=4$$", "result": "=4\\cdot\\:12" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:12=48$$", "result": "=48" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7kF09XBTycC+Pru1D11wnAQCWKUbvV6WK3fDUgFtg3Q/uZSIkOpnSKuqVg55rHK6o3sgi2dqxE8XWitN20kuh9of7CLCwPVYNCPoMKPEC9ZiwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=\\frac{3}{4}+48-15" }, { "type": "step", "primary": "Add/Subtract the numbers: $$48-15=33$$", "result": "=\\frac{3}{4}+33" }, { "type": "step", "primary": "Convert element to fraction: $$33=\\frac{33\\cdot\\:4}{4}$$", "result": "=\\frac{33\\cdot\\:4}{4}+\\frac{3}{4}" }, { "type": "step", "primary": "Since the denominators are equal, combine the fractions: $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{33\\cdot\\:4+3}{4}" }, { "type": "interim", "title": "$$33\\cdot\\:4+3=135$$", "input": "33\\cdot\\:4+3", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$33\\cdot\\:4=132$$", "result": "=132+3" }, { "type": "step", "primary": "Add the numbers: $$132+3=135$$", "result": "=135" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s72fuln4afzgHUiWDSchXGdVXTSum/z5kLpMzXS1UJIeyyyDX+pO7fRFf+fZoJn9Z+vdoWMdMNgk9z51N/sxYFcDkej7Yo2nDSDsL3ovikANwkt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "=\\frac{135}{4}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7d/Wv7FXVSokdBowymixWV7clJw3ekv7umsc4XCfDKsZK2mFTdh+fiTXnpim/sJz7wZKrlpcjsePZ3RFsQU5+oglAlm5MBjmOz6iqN1PySyW+lyZDB+4QT2KBTVDu0zODcTZnm/8YWra/1y/CNJ8TTB4pgUWEah0lniZLlD4X0wtkTA+s02x084jggHWovmKuzMaPB16cLAjbWseIFv44k8HRDkvNbcRhBR/OIrohgq2D/Ns1uYSWFhhjHWwpyqJ+" } }, { "type": "step", "result": "m=\\frac{135}{4}" } ], "meta": { "interimType": "General Equation Slope At Point 2Eq" } }, { "type": "interim", "title": "Find the line with slope m=$$\\frac{135}{4}$$ and passing through $$\\left(2,\\:\\frac{27}{2}\\right):{\\quad}y=\\frac{135}{4}x-54$$", "steps": [ { "type": "step", "primary": "Compute the line equation $$\\mathbf{y=mx+b}$$ for slope m=$$\\frac{135}{4}$$ and passing through $$\\left(2,\\:\\frac{27}{2}\\right)$$" }, { "type": "interim", "title": "Compute the $$y$$ intercept:$${\\quad}b=-54$$", "steps": [ { "type": "step", "primary": "Plug the slope $$\\frac{135}{4}$$ into $$y=mx+b$$", "result": "y=\\frac{135}{4}x+b" }, { "type": "step", "primary": "Plug in $$\\left(2,\\:\\frac{27}{2}\\right)$$: $$\\quad\\:x=2,\\:y=\\frac{27}{2}$$", "result": "\\frac{27}{2}=\\frac{135}{4}\\cdot\\:2+b" }, { "type": "step", "primary": "Isolate $$b$$" }, { "type": "interim", "title": "$$\\frac{27}{2}=\\frac{135}{4}\\cdot\\:2+b{\\quad:\\quad}b=-54$$", "input": "\\frac{27}{2}=\\frac{135}{4}\\cdot\\:2+b", "steps": [ { "type": "step", "primary": "Switch sides", "result": "\\frac{135}{4}\\cdot\\:2+b=\\frac{27}{2}" }, { "type": "interim", "title": "$$\\frac{135}{4}\\cdot\\:2=\\frac{135}{2}$$", "input": "\\frac{135}{4}\\cdot\\:2", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{135\\cdot\\:2}{4}" }, { "type": "step", "primary": "Multiply the numbers: $$135\\cdot\\:2=270$$", "result": "=\\frac{270}{4}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=\\frac{135}{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7llyDBAHTF9ew2CH9LL+2eonDdKucBi4LmQUhhh+vc191g99dC9fj9sg0EHzBIRDR0Avssn5BrwprqC36p8osjF5NkzKQgtswLlLi9MgL+goy2C7r05isu/GLHSmOSRIlB5FmSj1YOy9WAO788M9F2bNV4GUBpIL1Lgd4drm++RY=" } }, { "type": "step", "result": "\\frac{135}{2}+b=\\frac{27}{2}" }, { "type": "interim", "title": "Move $$\\frac{135}{2}\\:$$to the right side", "input": "\\frac{135}{2}+b=\\frac{27}{2}", "result": "b=-54", "steps": [ { "type": "step", "primary": "Subtract $$\\frac{135}{2}$$ from both sides", "result": "\\frac{135}{2}+b-\\frac{135}{2}=\\frac{27}{2}-\\frac{135}{2}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{135}{2}+b-\\frac{135}{2}=\\frac{27}{2}-\\frac{135}{2}", "result": "b=-54", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{135}{2}+b-\\frac{135}{2}:{\\quad}b$$", "input": "\\frac{135}{2}+b-\\frac{135}{2}", "steps": [ { "type": "step", "primary": "Add similar elements: $$\\frac{135}{2}-\\frac{135}{2}=0$$" }, { "type": "step", "result": "=b" } ], "meta": { "interimType": "Generic Simplify Specific 1Eq" } }, { "type": "interim", "title": "Simplify $$\\frac{27}{2}-\\frac{135}{2}:{\\quad}-54$$", "input": "\\frac{27}{2}-\\frac{135}{2}", "steps": [ { "type": "step", "primary": "Apply rule $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{27-135}{2}" }, { "type": "step", "primary": "Subtract the numbers: $$27-135=-108$$", "result": "=\\frac{-108}{2}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{108}{2}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{108}{2}=54$$", "result": "=-54" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7fABGtFkiImoa3ePQTLDtKc98Useq31EY8rcklFzhvhZV00rpv8+ZC6TM10tVCSHsFe/V7uhYFBoPiuAQQ+k0YKN6Hv6MoTMtvtU0IQwXdn9szOhN37mcRdV5CgGGkVwg2mShy4ch3wSZNr7gDxuUtHxAecG5HQFFiN5ZAbXAup8=" } }, { "type": "step", "result": "b=-54" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "result": "b=-54" } ], "meta": { "interimType": "Line Equation Find Intersection From Point 0Eq" } }, { "type": "step", "primary": "Construct the line equation $$\\mathbf{y=mx+b}$$ where $$\\mathbf{m}=\\frac{135}{4}$$ and $$\\mathbf{b}=-54$$", "result": "y=\\frac{135}{4}x-54" } ], "meta": { "interimType": "Line Equation Slope Point 6Eq" } }, { "type": "step", "result": "y=\\frac{135}{4}x-54" } ], "meta": { "solvingClass": "PreCalc" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "tangent y=4(x-\\frac{1}{x})^{3},\\at x=2" }, "showViewLarger": true } }, "meta": { "showVerify": true } }