tangent of f(x)=e^{-x}ln(x),\at x=1

{ "query": { "display": "tangent of $$f\\left(x\\right)=e^{-x}\\ln\\left(x\\right),\\:\\at\\:x=1$$", "symbolab_question": "PRE_CALC#tangent f(x)=e^{-x}\\ln(x),\\at x=1" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Derivative Applications", "subTopic": "Tangent", "default": "y=\\frac{1}{e}x-\\frac{1}{e}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Tangent line to $$f\\left(x\\right)=e^{-x}\\ln\\left(x\\right)$$, at $$x=1:{\\quad}y=\\frac{1}{e}x-\\frac{1}{e}$$", "steps": [ { "type": "interim", "title": "Find the tangent point:$${\\quad}\\left(1,\\:0\\right)$$", "steps": [ { "type": "step", "primary": "Plug $$x=1$$ into the equation $$f\\left(x\\right)=e^{-x}\\ln\\left(x\\right)$$", "result": "f\\left(x\\right)=e^{-1}\\ln\\left(1\\right)" }, { "type": "step", "primary": "Solve $$f\\left(x\\right)$$", "result": "f\\left(x\\right)=0" } ], "meta": { "interimType": "Tangent Find Tangent Point Title 0Eq" } }, { "type": "interim", "title": "Find the slope of $$f\\left(x\\right)=e^{-x}\\ln\\left(x\\right):{\\quad}\\frac{df\\left(x\\right)}{dx}=-e^{-x}\\ln\\left(x\\right)+\\frac{e^{-x}}{x}$$", "input": "f\\left(x\\right)=e^{-x}\\ln\\left(x\\right)", "steps": [ { "type": "step", "primary": "In order to find the slope of the function, take the derivative of $$e^{-x}\\ln\\left(x\\right)$$" }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(e^{-x}\\ln\\left(x\\right)\\right)=-e^{-x}\\ln\\left(x\\right)+\\frac{e^{-x}}{x}$$", "input": "\\frac{d}{dx}\\left(e^{-x}\\ln\\left(x\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the Product Rule: $$\\left(f{\\cdot}g\\right)'=f'{\\cdot}g+f{\\cdot}g'$$", "secondary": [ "$$f=e^{-x},\\:g=\\ln\\left(x\\right)$$" ], "result": "=\\frac{d}{dx}\\left(e^{-x}\\right)\\ln\\left(x\\right)+\\frac{d}{dx}\\left(\\ln\\left(x\\right)\\right)e^{-x}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Product%20Rule", "practiceTopic": "Product Rule" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(e^{-x}\\right)=-e^{-x}$$", "input": "\\frac{d}{dx}\\left(e^{-x}\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}e^{-x}\\frac{d}{dx}\\left(-x\\right)$$", "input": "\\frac{d}{dx}\\left(e^{-x}\\right)", "result": "=e^{-x}\\frac{d}{dx}\\left(-x\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=e^{u},\\:\\:u=-x$$" ], "result": "=\\frac{d}{du}\\left(e^{u}\\right)\\frac{d}{dx}\\left(-x\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(e^{u}\\right)=e^{u}$$", "input": "\\frac{d}{du}\\left(e^{u}\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{du}\\left(e^{u}\\right)=e^{u}$$", "result": "=e^{u}" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqCr3EWRZw3L4+rHTTdVG0Ok3hxk9aCfAWodBRxXgUexwx+RE9MtjN5hKMwTI7fffj/L0MoYg+CUn6oyL3EO7YrHahlpzKGY893KZ4T4i4Tv3RCXWsqiNx7T9zOhL5sYfw==" } }, { "type": "step", "result": "=e^{u}\\frac{d}{dx}\\left(-x\\right)" }, { "type": "step", "primary": "Substitute back $$u=-x$$", "result": "=e^{-x}\\frac{d}{dx}\\left(-x\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYmBepZeKV8bVrdHbYZEMXtOTdaV09PMxEKZ9FieghTFwc6p4sNpoW3XPzo2W3Ux6aosjLe8tD9HbrkG8vq6q9jiWR6DAIfSwNblE3ziU9D9b9KdAd3NQb43TJHJRafLwY/C30sSftAIFS6Qkpy19IkrNs0uRhmwmWtV92tk+c/8Zu/mDTHcAziAiYeNOzjloJg==" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(-x\\right)=-1$$", "input": "\\frac{d}{dx}\\left(-x\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=-\\frac{dx}{dx}" }, { "type": "step", "primary": "Apply the common derivative: $$\\frac{dx}{dx}=1$$", "result": "=-1" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYlIFiSG67zLdq9NRnof0E3fZGku9zFkxwe1dTH8vycb9+906N5fPsfVzu/TlcIEC4c1bIZxfodm3UsZcfZAZr4tZsYzC8RJ0SqIyNjewPF2dJLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=e^{-x}\\left(-1\\right)" }, { "type": "step", "primary": "Simplify", "result": "=-e^{-x}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(\\ln\\left(x\\right)\\right)=\\frac{1}{x}$$", "input": "\\frac{d}{dx}\\left(\\ln\\left(x\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dx}\\left(\\ln\\left(x\\right)\\right)=\\frac{1}{x}$$", "result": "=\\frac{1}{x}" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYhHxrkiFdmQgNsZN21633mEcjlLRK1jUV206qo4+vRN78rEus7TgCihQBF5omOFkJlc0OBMs8qTL4oWnxx62vyRTW26qciuyUBGXQExCUedYi3kiAkvXOTkrmcfV8WHLnF4CmnHjYZyazvJkuCAZs64=" } }, { "type": "step", "result": "=\\left(-e^{-x}\\right)\\ln\\left(x\\right)+\\frac{1}{x}e^{-x}" }, { "type": "interim", "title": "Simplify $$\\left(-e^{-x}\\right)\\ln\\left(x\\right)+\\frac{1}{x}e^{-x}:{\\quad}-e^{-x}\\ln\\left(x\\right)+\\frac{e^{-x}}{x}$$", "input": "\\left(-e^{-x}\\right)\\ln\\left(x\\right)+\\frac{1}{x}e^{-x}", "result": "=-e^{-x}\\ln\\left(x\\right)+\\frac{e^{-x}}{x}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=-e^{-x}\\ln\\left(x\\right)+\\frac{1}{x}e^{-x}" }, { "type": "interim", "title": "$$\\frac{1}{x}e^{-x}=\\frac{e^{-x}}{x}$$", "input": "\\frac{1}{x}e^{-x}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:e^{-x}}{x}" }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:e^{-x}=e^{-x}$$", "result": "=\\frac{e^{-x}}{x}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7oV/xLmpvbjXqT0r3Edcc2JGy01dqrQ5Rd9ZppWdx9aDMwViaLUXkeD+JukROhWdjo/xth/SjwoBLhkaMVoSR81roeUCC5gNxQc9h7CboxnNz6RsCHTbKW2tnTAM1c6hjdB3MQoH6daRfagfQ4GeaVfj03nA27Z0zjz3fi6FoIlw=" } }, { "type": "step", "result": "=-e^{-x}\\ln\\left(x\\right)+\\frac{e^{-x}}{x}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7f311KZ+vWnrT0a5BJdfiYqzrFjZStUzJzTcM5Dw5VuF6FOPbeU+fjQ/youbnJqb8cJChiVhDxT5N/LHSTLMjyFupciaP6co/POH3usZoTsC/FsgMdnQZXYwFwZTAsiGxWuh5QILmA3FBz2HsJujGcx4pgUWEah0lniZLlD4X0wtLtvZOJ29ks8MqISr1UWvcennBXpNGeS1NrFYNkAN/rlpoRsQrAqvLFyN8ZCKObJc=" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "-e^{-x}\\ln\\left(x\\right)+\\frac{e^{-x}}{x}" } ], "meta": { "interimType": "Slope Equation Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMGyxztx1DIZ4Y9QoeLjXWSwVhTonYBPVVOnmrCJXBr5Q45vezAXqyiEBFLUQXYhP4bkoH6MHGg0mtOsg27VY89g7ggd6D8X+XQG2x+mOeg3e+gGKY6KsK9lDkItldiGiO4gBJl4WMO1rA0a30/bUYlg==" } }, { "type": "interim", "title": "$$EN:\\:Title\\:General\\:Equation\\:Slope\\:At\\:Point\\:2Eq:{\\quad}m=\\frac{1}{e}$$", "steps": [ { "type": "step", "primary": "Plug $$x=1$$ into the equation $$-e^{-x}\\ln\\left(x\\right)+\\frac{e^{-x}}{x}$$", "result": "-e^{-1}\\ln\\left(1\\right)+\\frac{e^{-1}}{1}" }, { "type": "interim", "title": "Simplify $$-e^{-1}\\ln\\left(1\\right)+\\frac{e^{-1}}{1}:{\\quad}\\frac{1}{e}$$", "input": "-e^{-1}\\ln\\left(1\\right)+\\frac{e^{-1}}{1}", "result": "=\\frac{1}{e}", "steps": [ { "type": "step", "primary": "Apply rule $$\\frac{a}{1}=a$$", "secondary": [ "$$\\frac{e^{-1}}{1}=e^{-1}$$" ], "result": "=-e^{-1}\\ln\\left(1\\right)+e^{-1}" }, { "type": "interim", "title": "$$e^{-1}\\ln\\left(1\\right)=0$$", "input": "e^{-1}\\ln\\left(1\\right)", "steps": [ { "type": "interim", "title": "Simplify $$\\ln\\left(1\\right):{\\quad}0$$", "input": "\\ln\\left(1\\right)", "result": "=0\\cdot\\:e^{-1}", "steps": [ { "type": "step", "primary": "Apply log rule: $$\\log_a\\left(1\\right)=0$$", "result": "=0", "meta": { "practiceLink": "/practice/logarithms-practice", "practiceTopic": "Expand FOIL" } } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Apply rule $$0\\cdot\\:a=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ExISlLtcJp21c7LeXLGv6i061ljBSPJeENOw2efoSWtRZPRrfkNDmi+szkABFipUn8cp9RqO912RyV+lrdCBCc7RuMQ+KShx/NfZbf2mzeQ=" } }, { "type": "step", "result": "=-0+e^{-1}" }, { "type": "interim", "title": "Simplify", "input": "-0+e^{-1}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{-1}=\\frac{1}{a}$$", "secondary": [ "$$e^{-1}=\\frac{1}{e}$$" ], "result": "=-0+\\frac{1}{e}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "$$-0+\\frac{1}{e}=\\frac{1}{e}$$", "result": "=\\frac{1}{e}" } ], "meta": { "interimType": "Generic Simplify 0Eq" } }, { "type": "step", "result": "=\\frac{1}{e}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq" } }, { "type": "step", "result": "m=\\frac{1}{e}" } ], "meta": { "interimType": "General Equation Slope At Point 2Eq" } }, { "type": "interim", "title": "Find the line with slope m=$$\\frac{1}{e}$$ and passing through $$\\left(1,\\:0\\right):{\\quad}y=\\frac{1}{e}x-\\frac{1}{e}$$", "steps": [ { "type": "step", "primary": "Compute the line equation $$\\mathbf{y=mx+b}$$ for slope m=$$\\frac{1}{e}$$ and passing through $$\\left(1,\\:0\\right)$$" }, { "type": "interim", "title": "Compute the $$y$$ intercept:$${\\quad}b=-\\frac{1}{e}$$", "steps": [ { "type": "step", "primary": "Plug the slope $$\\frac{1}{e}$$ into $$y=mx+b$$", "result": "y=\\frac{1}{e}x+b" }, { "type": "step", "primary": "Plug in $$\\left(1,\\:0\\right)$$: $$\\quad\\:x=1,\\:y=0$$", "result": "0=\\frac{1}{e}\\cdot\\:1+b" }, { "type": "step", "primary": "Isolate $$b$$" }, { "type": "interim", "title": "$$0=\\frac{1}{e}\\cdot\\:1+b{\\quad:\\quad}b=-\\frac{1}{e}$$", "input": "0=\\frac{1}{e}\\cdot\\:1+b", "steps": [ { "type": "step", "primary": "Switch sides", "result": "\\frac{1}{e}\\cdot\\:1+b=0" }, { "type": "step", "primary": "Multiply: $$\\frac{1}{e}\\cdot\\:1=\\frac{1}{e}$$", "result": "\\frac{1}{e}+b=0" }, { "type": "interim", "title": "Move $$\\frac{1}{e}\\:$$to the right side", "input": "\\frac{1}{e}+b=0", "result": "b=-\\frac{1}{e}", "steps": [ { "type": "step", "primary": "Subtract $$\\frac{1}{e}$$ from both sides", "result": "\\frac{1}{e}+b-\\frac{1}{e}=0-\\frac{1}{e}" }, { "type": "step", "primary": "Simplify", "result": "b=-\\frac{1}{e}" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "result": "b=-\\frac{1}{e}" } ], "meta": { "interimType": "Line Equation Find Intersection From Point 0Eq" } }, { "type": "step", "primary": "Construct the line equation $$\\mathbf{y=mx+b}$$ where $$\\mathbf{m}=\\frac{1}{e}$$ and $$\\mathbf{b}=-\\frac{1}{e}$$", "result": "y=\\frac{1}{e}x-\\frac{1}{e}" } ], "meta": { "interimType": "Line Equation Slope Point 6Eq" } }, { "type": "step", "result": "y=\\frac{1}{e}x-\\frac{1}{e}" } ], "meta": { "solvingClass": "PreCalc" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "tangent f(x)=e^{-x}\\ln(x),\\at x=1" }, "showViewLarger": true } }, "meta": { "showVerify": true } }