derivative of y=x^x

{ "query": { "display": "derivative of $$y=x^{x}$$", "symbolab_question": "PRE_CALC#derivative y=x^{x}" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Derivatives", "subTopic": "Derivatives", "default": "x^{x}(\\ln(x)+1)", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{d}{dx}\\left(x^{x}\\right)=x^{x}\\left(\\ln\\left(x\\right)+1\\right)$$", "input": "\\frac{d}{dx}\\left(x^{x}\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b}=e^{b\\ln\\left(a\\right)}$$", "secondary": [ "$$x^{x}=e^{x\\ln\\left(x\\right)}$$" ], "result": "=\\frac{d}{dx}\\left(e^{x\\ln\\left(x\\right)}\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "Apply the chain rule:$${\\quad}e^{x\\ln\\left(x\\right)}\\frac{d}{dx}\\left(x\\ln\\left(x\\right)\\right)$$", "input": "\\frac{d}{dx}\\left(e^{x\\ln\\left(x\\right)}\\right)", "result": "=e^{x\\ln\\left(x\\right)}\\frac{d}{dx}\\left(x\\ln\\left(x\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=e^{u},\\:\\:u=x\\ln\\left(x\\right)$$" ], "result": "=\\frac{d}{du}\\left(e^{u}\\right)\\frac{d}{dx}\\left(x\\ln\\left(x\\right)\\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\\ln\\left(x\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=x\\ln\\left(x\\right)$$", "result": "=e^{x\\ln\\left(x\\right)}\\frac{d}{dx}\\left(x\\ln\\left(x\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYoDnFULkIYqxeOk/LudUfNHJZBliD2XWHJM352Ekucbz1NpEj4yUFTERoeqJRRLYHBPiZ+52xB2X1cQ6EdG5IQOcJIQC6oy+5f9LN8yyF1/zuHK7mNPD5TIt0sOO4YCLWRoVz+D5fLqtz7KfqHRLDSLvbBmbuQNTF0TphKZ8RuvabkCo9FlyJUsOgE5aAtC5rBG0MJHrQfXzSB22kHRiQ+E=" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(x\\ln\\left(x\\right)\\right)=\\ln\\left(x\\right)+1$$", "input": "\\frac{d}{dx}\\left(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": [ "$$f=x,\\:g=\\ln\\left(x\\right)$$" ], "result": "=\\frac{dx}{dx}\\ln\\left(x\\right)+\\frac{d}{dx}\\left(\\ln\\left(x\\right)\\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": "$$\\frac{d}{dx}\\left(\\ln\\left(x\\right)\\right)=\\frac{1}{x}$$", "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": "=1\\cdot\\:\\ln\\left(x\\right)+\\frac{1}{x}x" }, { "type": "interim", "title": "Simplify $$1\\cdot\\:\\ln\\left(x\\right)+\\frac{1}{x}x:{\\quad}\\ln\\left(x\\right)+1$$", "input": "1\\cdot\\:\\ln\\left(x\\right)+\\frac{1}{x}x", "result": "=\\ln\\left(x\\right)+1", "steps": [ { "type": "interim", "title": "$$1\\cdot\\:\\ln\\left(x\\right)=\\ln\\left(x\\right)$$", "input": "1\\cdot\\:\\ln\\left(x\\right)", "steps": [ { "type": "step", "primary": "Multiply: $$1\\cdot\\:\\ln\\left(x\\right)=\\ln\\left(x\\right)$$", "result": "=\\ln\\left(x\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79beA5k57NbkUABmP9cvSMNMvHyY50dhXPFfrjcmooUhCyVMMrE5H/on5k4a9Rxq6JLF4r9D4iCHLcmQJUoiTHPvjq12ZgmZKPkKiCTZgFKXH5WAyVHrJv05thO4WOh7+" } }, { "type": "interim", "title": "$$\\frac{1}{x}x=1$$", "input": "\\frac{1}{x}x", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:x}{x}" }, { "type": "step", "primary": "Cancel the common factor: $$x$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Vso/dWnUuor5ghlEsMXEfS061ljBSPJeENOw2efoSWs8auWUd4WNoosLPjGkhvjRR4IEq5gqBo0nbneAsjr1ThJsBb6oO/Vxx0S9zy9/5Xw=" } }, { "type": "step", "result": "=\\ln\\left(x\\right)+1" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79beA5k57NbkUABmP9cvSMD0ZkofBuWHItL2ASmW1plEtOtZYwUjyXhDTsNnn6ElrEWKbcJL5hTDKZvnQoIKbrNhc9kEQi6rnSOuqKtcWv6WLGmNnLPWGf9PH3lpmjoJIr32jhjKrBcQKNcH2dDElVRnODoV0IOvFDuxP4SwGLVrNWl2l6Tqo8vU+QPdaeikt" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=e^{x\\ln\\left(x\\right)}\\left(\\ln\\left(x\\right)+1\\right)" }, { "type": "interim", "title": "$$e^{x\\ln\\left(x\\right)}=x^{x}$$", "input": "e^{x\\ln\\left(x\\right)}", "result": "=x^{x}\\left(\\ln\\left(x\\right)+1\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{bc}=\\left(a^{b}\\right)^{c}$$", "result": "=\\left(e^{\\ln\\left(x\\right)}\\right)^{x}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Apply log rule: $$a^{\\log_{a}\\left(b\\right)}=b$$", "secondary": [ "$$e^{\\ln\\left(x\\right)}=x$$" ], "result": "=x^{x}", "meta": { "practiceLink": "/practice/logarithms-practice", "practiceTopic": "Expand FOIL" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7DPRjs78anA0615psKiy+hFXTSum/z5kLpMzXS1UJIexn4DmNH5jS5xc1x7lXmjV1P8vQyhiD4JSfqjIvcQ7tihQeggWMDKdNzq4N0nhKeNVr26dtsjdPi4JjJNhwGmrr" } } ], "meta": { "solvingClass": "Derivatives", "practiceLink": "/practice/derivatives-practice", "practiceTopic": "Derivatives" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=x^{x}(\\ln(x)+1)", "displayFormula": 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