(\partial)/(\partial x)((e^x)/(y+x^3))

{ "query": { "display": "$$\\frac{\\partial\\:}{\\partial\\:x}\\left(\\frac{e^{x}}{y+x^{3}}\\right)$$", "symbolab_question": "DERIVATIVE#\\frac{\\partial }{\\partial x}(\\frac{e^{x}}{y+x^{3}})" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Derivatives", "subTopic": "Partial Derivatives", "default": "\\frac{e^{x}(y+x^{3})-3x^{2}e^{x}}{(y+x^{3})^{2}}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:x}\\left(\\frac{e^{x}}{y+x^{3}}\\right)=\\frac{e^{x}\\left(y+x^{3}\\right)-3x^{2}e^{x}}{\\left(y+x^{3}\\right)^{2}}$$", "input": "\\frac{\\partial\\:}{\\partial\\:x}\\left(\\frac{e^{x}}{y+x^{3}}\\right)", "steps": [ { "type": "step", "primary": "Treat $$y\\:$$as a constant" }, { "type": "step", "primary": "Apply the Quotient Rule: $$\\left(\\frac{f}{g}\\right)'=\\frac{f'{\\cdot}g-g'{\\cdot}f}{g^{2}}$$", "result": "=\\frac{\\frac{\\partial\\:}{\\partial\\:x}\\left(e^{x}\\right)\\left(y+x^{3}\\right)-\\frac{\\partial\\:}{\\partial\\:x}\\left(y+x^{3}\\right)e^{x}}{\\left(y+x^{3}\\right)^{2}}" }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:x}\\left(e^{x}\\right)=e^{x}$$", "input": "\\frac{\\partial\\:}{\\partial\\:x}\\left(e^{x}\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{\\partial\\:}{\\partial\\:x}\\left(e^{x}\\right)=e^{x}$$", "result": "=e^{x}" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAnAZgtIsyiw2b31PDjA4hRyHI5S0StY1FdtOqqOPr0Te9fpuTWGA/bcx8V9AIGMLhha6HlAguYDcUHPYewm6MZzrzea8M9Hp5QtXq9EUDk+3vOlAMgng2wkuOmnhiWo9BEvf07Zwrc9Z9dExPFwr6bsJLd1ohke2Wgml78++2zI0g==" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:x}\\left(y+x^{3}\\right)=3x^{2}$$", "input": "\\frac{\\partial\\:}{\\partial\\:x}\\left(y+x^{3}\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{\\partial\\:}{\\partial\\:x}\\left(y\\right)+\\frac{\\partial\\:}{\\partial\\:x}\\left(x^{3}\\right)" }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:x}\\left(y\\right)=0$$", "input": "\\frac{\\partial\\:}{\\partial\\:x}\\left(y\\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+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAloe9eTz2HvO8muyM4Z1kKdlyEB4JYjIUjkjbDZ4tfSJ3+y6gfQnMr2Alg7BrHl9PbNWyGcX6HZt1LGXH2QGa+Ln0ClXHqmT3uOusVLMnE4CbAGGQy6xiZ+c/G1JYEHGagkt3WiGR7ZaCaXvz77bMjS" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:x}\\left(x^{3}\\right)=3x^{2}$$", "input": "\\frac{\\partial\\:}{\\partial\\:x}\\left(x^{3}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=3x^{3-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=3x^{2}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAnh+rWTho/HCR3Wl8+DsGSIHI5S0StY1FdtOqqOPr0Te8aEkN4RINY/GjMbKDxSWGpeTZMykILbMC5S4vTIC/oKMCpduEeI2njCEKkgMisPa5uBiDz/WIOhm2zsislrcVAGQhXG5F4pyGOhJT8nALgysIjaxJ4DvjTb2fbKjbvtlQ==" } }, { "type": "step", "result": "=0+3x^{2}" }, { "type": "step", "primary": "Simplify", "result": "=3x^{2}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{e^{x}\\left(y+x^{3}\\right)-3x^{2}e^{x}}{\\left(y+x^{3}\\right)^{2}}" } ], "meta": { "solvingClass": "Derivatives", "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Partial%20Derivatives", "practiceTopic": "Partial Derivatives" } }, "meta": { "showVerify": true } }