derivative of x^2e^{-3x}

{ "query": { "display": "derivative of $$x^{2}e^{-3x}$$", "symbolab_question": "PRE_CALC#derivative x^{2}e^{-3x}" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Derivatives", "subTopic": "Derivatives", "default": "2e^{-3x}x-3e^{-3x}x^{2}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{d}{dx}\\left(x^{2}e^{-3x}\\right)=2e^{-3x}x-3e^{-3x}x^{2}$$", "input": "\\frac{d}{dx}\\left(x^{2}e^{-3x}\\right)", "steps": [ { "type": "step", "primary": "Apply the Product Rule: $$\\left(f{\\cdot}g\\right)'=f'{\\cdot}g+f{\\cdot}g'$$", "secondary": [ "$$f=x^{2},\\:g=e^{-3x}$$" ], "result": "=\\frac{d}{dx}\\left(x^{2}\\right)e^{-3x}+\\frac{d}{dx}\\left(e^{-3x}\\right)x^{2}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Product%20Rule", "practiceTopic": "Product Rule" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(x^{2}\\right)=2x$$", "input": "\\frac{d}{dx}\\left(x^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2x^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYkmb3s5xAUYje7fZWSRkdb2k3hxk9aCfAWodBRxXgUexcQsmN/cITrVSOMImEqe3fkeCBKuYKgaNJ253gLI69U7cjrVUqImvoUuRtb+2ccCzWsr9JoDNJaP7hueshcYJ6w==" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(e^{-3x}\\right)=-3e^{-3x}$$", "input": "\\frac{d}{dx}\\left(e^{-3x}\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}e^{-3x}\\frac{d}{dx}\\left(-3x\\right)$$", "input": "\\frac{d}{dx}\\left(e^{-3x}\\right)", "result": "=e^{-3x}\\frac{d}{dx}\\left(-3x\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=e^{u},\\:\\:u=-3x$$" ], "result": "=\\frac{d}{du}\\left(e^{u}\\right)\\frac{d}{dx}\\left(-3x\\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(-3x\\right)" }, { "type": "step", "primary": "Substitute back $$u=-3x$$", "result": "=e^{-3x}\\frac{d}{dx}\\left(-3x\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYo4Zr3/RPCZRiO/XTUD1MnQcjlLRK1jUV206qo4+vRN797KvhEOUentkJU3Q6unc+igIFotFu6oyM0XWl+PP5OmOqu257D3ZahvGN/u/p/wtgmDcorQdvxjXaqJSC3h950UqTd96MWTKI6Kr2Ib0iQBbgTZHtHmQDZkDHT6lbFNT6vT1FATJsc8ZttpCA6/NQQ==" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(-3x\\right)=-3$$", "input": "\\frac{d}{dx}\\left(-3x\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=-3\\frac{dx}{dx}" }, { "type": "step", "primary": "Apply the common derivative: $$\\frac{dx}{dx}=1$$", "result": "=-3\\cdot\\:1" }, { "type": "step", "primary": "Simplify", "result": "=-3", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYlcDxdwCpciGk7RgERTTuG+XIQHgliMhSOSNsNni19In+hOS6pvnfiLEcPgP/EgopA4bfwiV6iMLJ5sC1nL7dObQVTsDMCQjIcyOcFlJ7z1QsIjaxJ4DvjTb2fbKjbvtlQ==" } }, { "type": "step", "result": "=e^{-3x}\\left(-3\\right)" }, { "type": "step", "primary": "Simplify", "result": "=-3e^{-3x}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=2xe^{-3x}+\\left(-3e^{-3x}\\right)x^{2}" }, { "type": "step", "primary": "Simplify", "result": "=2e^{-3x}x-3e^{-3x}x^{2}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "practiceLink": "/practice/derivatives-practice", "practiceTopic": "Derivatives" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "y=2e^{-3x}x-3e^{-3x}x^{2}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }