derivative of y=x^3ln(x)

{ "query": { "display": "derivative of $$y=x^{3}\\ln\\left(x\\right)$$", "symbolab_question": "PRE_CALC#derivative y=x^{3}\\ln(x)" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Derivatives", "subTopic": "Derivatives", "default": "3x^{2}\\ln(x)+x^{2}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{d}{dx}\\left(x^{3}\\ln\\left(x\\right)\\right)=3x^{2}\\ln\\left(x\\right)+x^{2}$$", "input": "\\frac{d}{dx}\\left(x^{3}\\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": [ "$$f=x^{3},\\:g=\\ln\\left(x\\right)$$" ], "result": "=\\frac{d}{dx}\\left(x^{3}\\right)\\ln\\left(x\\right)+\\frac{d}{dx}\\left(\\ln\\left(x\\right)\\right)x^{3}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Product%20Rule", "practiceTopic": "Product Rule" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(x^{3}\\right)=3x^{2}$$", "input": "\\frac{d}{dx}\\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+idP5vLVrjUll6eMdYtb6j95rHG7YtZ73Xx2qCjqk3hxk9aCfAWodBRxXgUexf7nh0v5ML3fMP9GgRVbRX/8//6/nV5O4fb8Xgwi7maqO95Is7YBdcRqXmTH8euc2MRY7LLv3QukQErzdJ9wdtQ==" } }, { "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": "=3x^{2}\\ln\\left(x\\right)+\\frac{1}{x}x^{3}" }, { "type": "interim", "title": "$$\\frac{1}{x}x^{3}=x^{2}$$", "input": "\\frac{1}{x}x^{3}", "result": "=3x^{2}\\ln\\left(x\\right)+x^{2}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:x^{3}}{x}" }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:x^{3}=x^{3}$$", "result": "=\\frac{x^{3}}{x}" }, { "type": "step", "primary": "Cancel the common factor: $$x$$", "result": "=x^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7WVLhIKJxAHj/60V2C9gzaxZsnww+P7XjdokU4IcmGg+jkVi15I8rBefLi4Iyt2wrDzRG5i2RnXJK7EZvbhYwA2yKJewY067QC1vSYz7l2iHTFRig/D9Ntfz6335WeeLk" } } ], "meta": { "solvingClass": "Derivatives", "practiceLink": "/practice/derivatives-practice", "practiceTopic": "Derivatives" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=3x^{2}\\ln(x)+x^{2}", "displayFormula": "y=3x^{2}\\ln(x)+x^{2}", "derivativeFormula": "x(6\\ln(x)+5)", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false }, "xmin": 0, "calculatePoints": true } ] }, "pointsToDraw": { "pointsLatex": [ "(0,0)", "(\\frac{1}{\\sqrt[3]{e}},0)", "(\\frac{1}{e^{\\frac{5}{6}}},-\\frac{3}{2e^{\\frac{5}{3}}})" ], "pointsDecimal": [ { "fst": 0, "snd": 0 }, { "fst": 0.7165313105737893, "snd": 0 }, { "fst": 0.4345982085070782, "snd": -0.28331340425634277 } ], "attributes": [ { "color": "white", "borderColor": "PURPLE", "labels": [ "\\mathrm{Undefined}" ], "labelTypes": [ "DEFAULT" ], "labelColors": [ "PURPLE" ] }, { "color": "BLUE", "labels": [ "\\mathrm{x-intercept}" ], "labelTypes": [ "INTERCEPT" ], "labelColors": [ "BLUE" ] }, { "color": "BLUE", "labels": [ "\\mathrm{Global\\:Minimum}" ], "labelTypes": [ "EXTREME" ], "labelColors": [ "BLUE" ] } ] }, "functionChanges": [ { "origFormulaLatex": [], "finalFormulaLatex": [], "plotTitle": "y=3x^{2}\\ln(x)+x^{2}", "paramsLatex": [], "paramsReplacementsLatex": [] } ], "localBoundingBox": { "xMin": -2.0894053016331693, "xMax": 2.7342834811495793, "yMin": -0.80534668293908, "yMax": 0.5531977531509348 } }, "showViewLarger": true } }, "meta": { "showVerify": true } }