(\partial)/(\partial x)(cos(xy^2))

{ "query": { "display": "$$\\frac{\\partial\\:}{\\partial\\:x}\\left(\\cos\\left(xy^{2}\\right)\\right)$$", "symbolab_question": "DERIVATIVE#\\frac{\\partial }{\\partial x}(\\cos(xy^{2}))" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Derivatives", "subTopic": "Partial Derivatives", "default": "-\\sin(xy^{2})y^{2}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:x}\\left(\\cos\\left(xy^{2}\\right)\\right)=-\\sin\\left(xy^{2}\\right)y^{2}$$", "input": "\\frac{\\partial\\:}{\\partial\\:x}\\left(\\cos\\left(xy^{2}\\right)\\right)", "steps": [ { "type": "step", "primary": "Treat $$y\\:$$as a constant" }, { "type": "interim", "title": "Apply the chain rule:$${\\quad}-\\sin\\left(xy^{2}\\right)\\frac{\\partial\\:}{\\partial\\:x}\\left(xy^{2}\\right)$$", "input": "\\frac{\\partial\\:}{\\partial\\:x}\\left(\\cos\\left(xy^{2}\\right)\\right)", "result": "=-\\sin\\left(xy^{2}\\right)\\frac{\\partial\\:}{\\partial\\:x}\\left(xy^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=\\cos\\left(u\\right),\\:\\:u=xy^{2}$$" ], "result": "=\\frac{\\partial\\:}{\\partial\\:u}\\left(\\cos\\left(u\\right)\\right)\\frac{\\partial\\:}{\\partial\\:x}\\left(xy^{2}\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:u}\\left(\\cos\\left(u\\right)\\right)=-\\sin\\left(u\\right)$$", "input": "\\frac{\\partial\\:}{\\partial\\:u}\\left(\\cos\\left(u\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{\\partial\\:}{\\partial\\:u}\\left(\\cos\\left(u\\right)\\right)=-\\sin\\left(u\\right)$$", "result": "=-\\sin\\left(u\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAncc9yOeUX2aanUOtw4oGI+MefbjLlytLm8i7rCBwAkD6ORWLXkjysF58uLgjK3bCu8ULOfUcA4pVZWisIOgp2IU1tuqnIrslARl0BMQlHnWDzoUJBDlYJcZFFoXZRVFxPHj7Po+YX+DlGxZkQ+rAL6Y+HaAg3bsYzGEwLhAhyh4w==" } }, { "type": "step", "result": "=-\\sin\\left(u\\right)\\frac{\\partial\\:}{\\partial\\:x}\\left(xy^{2}\\right)" }, { "type": "step", "primary": "Substitute back $$u=xy^{2}$$", "result": "=-\\sin\\left(xy^{2}\\right)\\frac{\\partial\\:}{\\partial\\:x}\\left(xy^{2}\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAlCpWEplKctzmXtYs3QJVtvh5/nV8AGlYUAqApumu7Dg2dxqBuiqPIwz4eL8pobNPrZb/GFTMIf3X+UjAHwJqSVyYNYEDGka44qNIndOOZXMQfskHYt3pVbumpa7hKQ4cv0mLAQHRleZoyoxY02lsKuzxqs0FQn7NY3QGcmDxv+AenWDmmbtV2WXTCxOG/pES/dKZS9fj5nGO8F191IYKw7ClwHVv5rKpMbWSxPD5Uhrr8yD3hLQ33B7/8/LpbPE3o=" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:x}\\left(xy^{2}\\right)=y^{2}$$", "input": "\\frac{\\partial\\:}{\\partial\\:x}\\left(xy^{2}\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=y^{2}\\frac{\\partial\\:}{\\partial\\:x}\\left(x\\right)" }, { "type": "step", "primary": "Apply the common derivative: $$\\frac{\\partial\\:}{\\partial\\:x}\\left(x\\right)=1$$", "result": "=y^{2}\\cdot\\:1" }, { "type": "step", "primary": "Simplify", "result": "=y^{2}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAk17pAcKx/7g8NNuOxBDtHB/M3hESAhJ79Q7OQNfwXSMPtUKyVYYGMaaXpbskiz/95eTZMykILbMC5S4vTIC/oKMCpduEeI2njCEKkgMisPa5uBiDz/WIOhm2zsislrcVD4RMzl+mOb0hHBObNWZnY1sIjaxJ4DvjTb2fbKjbvtlQ==" } }, { "type": "step", "result": "=-\\sin\\left(xy^{2}\\right)y^{2}" } ], "meta": { "solvingClass": "Derivatives", "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Partial%20Derivatives", "practiceTopic": "Partial Derivatives" } }, "meta": { "showVerify": true } }