derivative of y=x^{ln(x)}

{ "query": { "display": "derivative of $$y=x^{\\ln\\left(x\\right)}$$", "symbolab_question": "PRE_CALC#derivative y=x^{\\ln(x)}" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Derivatives", "subTopic": "Derivatives", "default": "\\frac{2e^{\\ln^{2}(x)}\\ln(x)}{x}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{d}{dx}\\left(x^{\\ln\\left(x\\right)}\\right)=\\frac{2e^{\\ln^{2}\\left(x\\right)}\\ln\\left(x\\right)}{x}$$", "input": "\\frac{d}{dx}\\left(x^{\\ln\\left(x\\right)}\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b}=e^{b\\ln\\left(a\\right)}$$", "secondary": [ "$$x^{\\ln\\left(x\\right)}=e^{\\ln\\left(x\\right)\\ln\\left(x\\right)}$$" ], "result": "=\\frac{d}{dx}\\left(e^{\\ln\\left(x\\right)\\ln\\left(x\\right)}\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "Apply the chain rule:$${\\quad}e^{\\ln\\left(x\\right)\\ln\\left(x\\right)}\\frac{d}{dx}\\left(\\ln\\left(x\\right)\\ln\\left(x\\right)\\right)$$", "input": "\\frac{d}{dx}\\left(e^{\\ln\\left(x\\right)\\ln\\left(x\\right)}\\right)", "result": "=e^{\\ln\\left(x\\right)\\ln\\left(x\\right)}\\frac{d}{dx}\\left(\\ln\\left(x\\right)\\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": [ "$$f=e^{u},\\:\\:u=\\ln\\left(x\\right)\\ln\\left(x\\right)$$" ], "result": "=\\frac{d}{du}\\left(e^{u}\\right)\\frac{d}{dx}\\left(\\ln\\left(x\\right)\\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(\\ln\\left(x\\right)\\ln\\left(x\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\ln\\left(x\\right)\\ln\\left(x\\right)$$", "result": "=e^{\\ln\\left(x\\right)\\ln\\left(x\\right)}\\frac{d}{dx}\\left(\\ln\\left(x\\right)\\ln\\left(x\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYuQq+CPtDr49VTgnQE594bxTAVcBTGqIKB45P6vQbxYzLI71+ylVDvwHghUiHeEt94lsPYObl85kuSl+cyMzh2V4l9+XUP4dNZSL8WpmbbeTMnd0xCSi36QkrccGK1KENGRgCUIJEdloNKL77khvysnk0Paq7AMvVYpcRbo+YvW1o3oe/oyhMy2+1TQhDBd2fzGrMGuaCGoCkBTn65ypufhm2/z8V92sLreSsUccmX/A" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(\\ln\\left(x\\right)\\ln\\left(x\\right)\\right)=\\frac{2\\ln\\left(x\\right)}{x}$$", "input": "\\frac{d}{dx}\\left(\\ln\\left(x\\right)\\ln\\left(x\\right)\\right)", "steps": [ { "type": "interim", "title": "Simplify $$\\ln\\left(x\\right)\\ln\\left(x\\right):{\\quad}\\ln^{2}\\left(x\\right)$$", "input": "\\ln\\left(x\\right)\\ln\\left(x\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\ln\\left(x\\right)\\ln\\left(x\\right)=\\:\\ln^{1+1}\\left(x\\right)$$" ], "result": "=\\ln^{1+1}\\left(x\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=\\ln^{2}\\left(x\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\frac{d}{dx}\\left(\\ln^{2}\\left(x\\right)\\right)" }, { "type": "interim", "title": "Apply the chain rule:$${\\quad}2\\ln\\left(x\\right)\\frac{d}{dx}\\left(\\ln\\left(x\\right)\\right)$$", "input": "\\frac{d}{dx}\\left(\\ln^{2}\\left(x\\right)\\right)", "result": "=2\\ln\\left(x\\right)\\frac{d}{dx}\\left(\\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": [ "$$f=u^{2},\\:\\:u=\\ln\\left(x\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{2}\\right)\\frac{d}{dx}\\left(\\ln\\left(x\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{2}\\right)=2u$$", "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}{dx}\\left(\\ln\\left(x\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\ln\\left(x\\right)$$", "result": "=2\\ln\\left(x\\right)\\frac{d}{dx}\\left(\\ln\\left(x\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYryBTlMeFYklLS+7NGY7IrbgLLGDwG8USUPGEfybPT5wdLl7DeVd7l7l/uUT/v1GhNJLZOXSTCXkMFKW90A2Pi9LK+y6MPtYTdJvOARgtNbn9u0fqgJ8zpSyZd7hSBwk/xytPSdOiOs3GnHEr/6tFrGqkIX6kseCKdEth+cILnwyMX9qYKk0viGvXKHsQSVFeoi+ZPlQ6KBam0kIORBN9aU=" } }, { "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": "=2\\ln\\left(x\\right)\\frac{1}{x}" }, { "type": "interim", "title": "Simplify $$2\\ln\\left(x\\right)\\frac{1}{x}:{\\quad}\\frac{2\\ln\\left(x\\right)}{x}$$", "input": "2\\ln\\left(x\\right)\\frac{1}{x}", "result": "=\\frac{2\\ln\\left(x\\right)}{x}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:2\\ln\\left(x\\right)}{x}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:2=2$$", "result": "=\\frac{2\\ln\\left(x\\right)}{x}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ssfCJL4wAEgWAIEELhXhVWFOMt0eyUHA21ffJamMj/erju+5Z51e/ZZSD3gRHwjBciTxtIXYwJ1XFysJkI4fizUmq4EyRR8dGqLVVLtyMoIeNvb7k0sVmuwf19w9aD9NPyj+WkMythK9g49524vLkvmZlIj8NEhDZfF+u7FP2E4=" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=e^{\\ln\\left(x\\right)\\ln\\left(x\\right)}\\frac{2\\ln\\left(x\\right)}{x}" }, { "type": "interim", "title": "Simplify $$e^{\\ln\\left(x\\right)\\ln\\left(x\\right)}\\frac{2\\ln\\left(x\\right)}{x}:{\\quad}\\frac{2e^{\\ln^{2}\\left(x\\right)}\\ln\\left(x\\right)}{x}$$", "input": "e^{\\ln\\left(x\\right)\\ln\\left(x\\right)}\\frac{2\\ln\\left(x\\right)}{x}", "result": "=\\frac{2e^{\\ln^{2}\\left(x\\right)}\\ln\\left(x\\right)}{x}", "steps": [ { "type": "interim", "title": "$$e^{\\ln\\left(x\\right)\\ln\\left(x\\right)}=e^{\\ln^{2}\\left(x\\right)}$$", "input": "e^{\\ln\\left(x\\right)\\ln\\left(x\\right)}", "steps": [ { "type": "interim", "title": "Simplify $$\\ln\\left(x\\right)\\ln\\left(x\\right):{\\quad}\\ln^{2}\\left(x\\right)$$", "input": "\\ln\\left(x\\right)\\ln\\left(x\\right)", "result": "=e^{\\ln^{2}\\left(x\\right)}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\ln\\left(x\\right)\\ln\\left(x\\right)=\\:\\ln^{1+1}\\left(x\\right)$$" ], "result": "=\\ln^{1+1}\\left(x\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=\\ln^{2}\\left(x\\right)" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Wp3zOl63Ds2i1eDBnB3VeTyH67l9XiIECQQI/Jd/snnMwViaLUXkeD+JukROhWdjLGx/xPUYqw4McBoyx9VYmD/L0MoYg+CUn6oyL3EO7Yoyd3TEJKLfpCStxwYrUoQ0ElREGpIvId0gpKCxPtlWPDAJzpXYgVu2y16nzD1HI8o=" } }, { "type": "step", "result": "=e^{\\ln^{2}\\left(x\\right)}\\frac{2\\ln\\left(x\\right)}{x}" }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{2\\ln\\left(x\\right)e^{\\ln^{2}\\left(x\\right)}}{x}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Wp3zOl63Ds2i1eDBnB3VeYJ5OsvjQ4nSL+yHDxcXBiRy3mheBdVORXEhuyjAM41WCUCWbkwGOY7PqKo3U/JLJcMY6In0TUYoV+ph91E/gm648pZh5HU+F96kzShySXIRvuuMNoCRXE40bWKEFclVjx429vuTSxWa7B/X3D1oP018UMc65xVcf5uIYUJty8cMdc5IpcUhqAlilNAjUoM2B5Bkq5LvR/hgxj0pdsBkeRMkt3WiGR7ZaCaXvz77bMjS" } } ], "meta": { "solvingClass": "Derivatives", 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