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

{ "query": { "display": "derivative of $$y=x^{\\sin\\left(x\\right)}$$", "symbolab_question": "PRE_CALC#derivative y=x^{\\sin(x)}" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Derivatives", "subTopic": "Derivatives", "default": "x^{\\sin(x)}(\\cos(x)\\ln(x)+\\frac{\\sin(x)}{x})", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{d}{dx}\\left(x^{\\sin\\left(x\\right)}\\right)=x^{\\sin\\left(x\\right)}\\left(\\cos\\left(x\\right)\\ln\\left(x\\right)+\\frac{\\sin\\left(x\\right)}{x}\\right)$$", "input": "\\frac{d}{dx}\\left(x^{\\sin\\left(x\\right)}\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b}=e^{b\\ln\\left(a\\right)}$$", "secondary": [ "$$x^{\\sin\\left(x\\right)}=e^{\\sin\\left(x\\right)\\ln\\left(x\\right)}$$" ], "result": "=\\frac{d}{dx}\\left(e^{\\sin\\left(x\\right)\\ln\\left(x\\right)}\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "Apply the chain rule:$${\\quad}e^{\\sin\\left(x\\right)\\ln\\left(x\\right)}\\frac{d}{dx}\\left(\\sin\\left(x\\right)\\ln\\left(x\\right)\\right)$$", "input": "\\frac{d}{dx}\\left(e^{\\sin\\left(x\\right)\\ln\\left(x\\right)}\\right)", "result": "=e^{\\sin\\left(x\\right)\\ln\\left(x\\right)}\\frac{d}{dx}\\left(\\sin\\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": [ "$$f=e^{u},\\:\\:u=\\sin\\left(x\\right)\\ln\\left(x\\right)$$" ], "result": "=\\frac{d}{du}\\left(e^{u}\\right)\\frac{d}{dx}\\left(\\sin\\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(\\sin\\left(x\\right)\\ln\\left(x\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sin\\left(x\\right)\\ln\\left(x\\right)$$", "result": "=e^{\\sin\\left(x\\right)\\ln\\left(x\\right)}\\frac{d}{dx}\\left(\\sin\\left(x\\right)\\ln\\left(x\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYsGApDRoFf+VLMrgxruGQlqKdN/E2jtb+g0YTDJY8PUBK576WENGsoCX8pronHaqWbnEbM1x5wGAstoUG+Ob1j8Zqb5UjmNAHi9A1q3z5bAugS7T18bJL5ijEKR5uwoyaDHWt34JKIzMe6bg40b5bVSdM/tmThqxSSpAg04Ln0AUVLYe+dYgmxE+FkN3sPupDd0plL1+PmcY7wXX3UhgrDsKXAdW/msqkxtZLE8PlSGuvzIPeEtDfcHv/z8uls8Teg==" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(\\sin\\left(x\\right)\\ln\\left(x\\right)\\right)=\\cos\\left(x\\right)\\ln\\left(x\\right)+\\frac{\\sin\\left(x\\right)}{x}$$", "input": "\\frac{d}{dx}\\left(\\sin\\left(x\\right)\\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": [ "$$f=\\sin\\left(x\\right),\\:g=\\ln\\left(x\\right)$$" ], "result": "=\\frac{d}{dx}\\left(\\sin\\left(x\\right)\\right)\\ln\\left(x\\right)+\\frac{d}{dx}\\left(\\ln\\left(x\\right)\\right)\\sin\\left(x\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Product%20Rule", "practiceTopic": "Product Rule" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(\\sin\\left(x\\right)\\right)=\\cos\\left(x\\right)$$", "input": "\\frac{d}{dx}\\left(\\sin\\left(x\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dx}\\left(\\sin\\left(x\\right)\\right)=\\cos\\left(x\\right)$$", "result": "=\\cos\\left(x\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYgOt2FhQQwx0GxLGzv2mPOv8zeERICEnv1Ds5A1/BdIwQslTDKxOR/6J+ZOGvUcaugB66mSUqneplfTkjggryzA+YUX37Aa/AAEf1Hkty8FUj7LPbFLewMJWlj8VtjhXr5J/4xg9Nn6C/zrAXreziPc=" } }, { "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": "=\\cos\\left(x\\right)\\ln\\left(x\\right)+\\frac{1}{x}\\sin\\left(x\\right)" }, { "type": "interim", "title": "$$\\frac{1}{x}\\sin\\left(x\\right)=\\frac{\\sin\\left(x\\right)}{x}$$", "input": "\\frac{1}{x}\\sin\\left(x\\right)", "result": "=\\cos\\left(x\\right)\\ln\\left(x\\right)+\\frac{\\sin\\left(x\\right)}{x}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:\\sin\\left(x\\right)}{x}" }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:\\sin\\left(x\\right)=\\sin\\left(x\\right)$$", "result": "=\\frac{\\sin\\left(x\\right)}{x}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7GrAJyMm9EutN1XxUWBlsk0UoSFq5qFTVZ48q18DDBn+rju+5Z51e/ZZSD3gRHwjBfD9S8n6nKr+eQTaGvs78rzUmq4EyRR8dGqLVVLtyMoLf/4B3/1Zs1Z0//Np9rBD3g4tTf5kr07YcyabJuQwy/L3gBTAzzse3WNna8JAh1Y4kt3WiGR7ZaCaXvz77bMjS" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=e^{\\sin\\left(x\\right)\\ln\\left(x\\right)}\\left(\\cos\\left(x\\right)\\ln\\left(x\\right)+\\frac{\\sin\\left(x\\right)}{x}\\right)" }, { "type": "interim", "title": "$$e^{\\sin\\left(x\\right)\\ln\\left(x\\right)}=x^{\\sin\\left(x\\right)}$$", "input": "e^{\\sin\\left(x\\right)\\ln\\left(x\\right)}", "result": "=x^{\\sin\\left(x\\right)}\\left(\\cos\\left(x\\right)\\ln\\left(x\\right)+\\frac{\\sin\\left(x\\right)}{x}\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{bc}=\\left(a^{b}\\right)^{c}$$", "result": "=\\left(e^{\\ln\\left(x\\right)}\\right)^{\\sin\\left(x\\right)}", "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^{\\sin\\left(x\\right)}", "meta": { "practiceLink": "/practice/logarithms-practice", "practiceTopic": "Expand FOIL" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7+zdbgvF7Kvq8KXk+Ly0G19DBO5hrU4N/Y4DzXzU8pq5wkKGJWEPFPk38sdJMsyPI7Og3QwZoPKXLf5qD8epwh2nKyrRYyuQTMNSmQlKKcsNwnm4MnRE4fFy045FZw5BtTdXAxAEcIet//rB/LdIvhCS3daIZHtloJpe/PvtsyNI=" } } ], "meta": { "solvingClass": "Derivatives", "practiceLink": "/practice/derivatives-practice", "practiceTopic": "Derivatives" } }, 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