(\partial)/(\partial y)(z^2x+y^3)

{ "query": { "display": "$$\\frac{\\partial\\:}{\\partial\\:y}\\left(z^{2}x+y^{3}\\right)$$", "symbolab_question": "DERIVATIVE#\\frac{\\partial }{\\partial y}(z^{2}x+y^{3})" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Derivatives", "subTopic": "Partial Derivatives", "default": "3y^{2}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:y}\\left(z^{2}x+y^{3}\\right)=3y^{2}$$", "input": "\\frac{\\partial\\:}{\\partial\\:y}\\left(z^{2}x+y^{3}\\right)", "steps": [ { "type": "step", "primary": "Treat $$z,\\:x\\:$$as constants" }, { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{\\partial\\:}{\\partial\\:y}\\left(z^{2}x\\right)+\\frac{\\partial\\:}{\\partial\\:y}\\left(y^{3}\\right)" }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:y}\\left(z^{2}x\\right)=0$$", "input": "\\frac{\\partial\\:}{\\partial\\:y}\\left(z^{2}x\\right)", "steps": [ { "type": "step", "primary": "Derivative of a constant: $$\\frac{d}{dx}\\left({a}\\right)=0$$", "result": "=0" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAnEYUCRvHeUUOFt75DckgBY/M3hESAhJ79Q7OQNfwXSMCQINjQkh1y9irIetDOTsvVkS3dlcCKpQTQcheuut7Mkm+hmRJA1ZPgdMDAPJn089r4HT6P72DJc5g5UU8VRQVXTa8YR7c3+IUKUvsfm5YH9" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:y}\\left(y^{3}\\right)=3y^{2}$$", "input": "\\frac{\\partial\\:}{\\partial\\:y}\\left(y^{3}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=3y^{3-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=3y^{2}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAkvrdCcXEGzAPx3zbZxXj+vHI5S0StY1FdtOqqOPr0Te/n/d//j7SBxmh3u6ywY5pleTZMykILbMC5S4vTIC/oKMCpduEeI2njCEKkgMisPa5uBiDz/WIOhm2zsislrcVB/Ni+Upjp2PaifAWzeMC+xsIjaxJ4DvjTb2fbKjbvtlQ==" } }, { "type": "step", "result": "=0+3y^{2}" }, { "type": "step", "primary": "Simplify", "result": "=3y^{2}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Partial%20Derivatives", "practiceTopic": "Partial Derivatives" } }, "meta": { "showVerify": true } }