derivative of 2x*e^{x^2}

{ "query": { "display": "derivative of $$2x\\cdot\\:e^{x^{2}}$$", "symbolab_question": "PRE_CALC#derivative 2x\\cdot e^{x^{2}}" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Derivatives", "subTopic": "Derivatives", "default": "2(e^{x^{2}}+2e^{x^{2}}x^{2})", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{d}{dx}\\left(2xe^{x^{2}}\\right)=2\\left(e^{x^{2}}+2e^{x^{2}}x^{2}\\right)$$", "input": "\\frac{d}{dx}\\left(2xe^{x^{2}}\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=2\\frac{d}{dx}\\left(xe^{x^{2}}\\right)" }, { "type": "step", "primary": "Apply the Product Rule: $$\\left(f{\\cdot}g\\right)'=f'{\\cdot}g+f{\\cdot}g'$$", "secondary": [ "$$f=x,\\:g=e^{x^{2}}$$" ], "result": "=2\\left(\\frac{dx}{dx}e^{x^{2}}+\\frac{d}{dx}\\left(e^{x^{2}}\\right)x\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Product%20Rule", "practiceTopic": "Product Rule" } }, { "type": "interim", "title": "$$\\frac{dx}{dx}=1$$", "input": "\\frac{dx}{dx}", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{dx}{dx}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYko/29fz701XcRtz4b42RqRjqLYrB3CcI0Y7zGHBJCja+8ZDu8iF4MSewt4yms1lIdz2XHFZ6BxfaHSMA6lT+lbVmoiKRd+ttkZ9NIrGodT+" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(e^{x^{2}}\\right)=e^{x^{2}}\\cdot\\:2x$$", "input": "\\frac{d}{dx}\\left(e^{x^{2}}\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}e^{x^{2}}\\frac{d}{dx}\\left(x^{2}\\right)$$", "input": "\\frac{d}{dx}\\left(e^{x^{2}}\\right)", "result": "=e^{x^{2}}\\frac{d}{dx}\\left(x^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=e^{u},\\:\\:u=x^{2}$$" ], "result": "=\\frac{d}{du}\\left(e^{u}\\right)\\frac{d}{dx}\\left(x^{2}\\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(x^{2}\\right)" }, { "type": "step", "primary": "Substitute back $$u=x^{2}$$", "result": "=e^{x^{2}}\\frac{d}{dx}\\left(x^{2}\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYtEScpZXYhfWKLYA8boZzOzCRJgoAMiTd6o06hbXd0s3OK1n91tyBoBr/ZHP0eNC/RSNU68ZmiYZN//Vg53tMEy0onLoy5qG/jxJORJuOa69tM+8i4MfCTfnvkTDikjRz9+aMILno9fGR346bANfmIgEuDOVaQvKofqHoY5jNapszx1SRmkdwriKeTV96yww0ImpXFf3SOUx+H18qfp3MLg=" } }, { "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": "step", "result": "=e^{x^{2}}\\cdot\\:2x" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=2\\left(1\\cdot\\:e^{x^{2}}+e^{x^{2}}\\cdot\\:2xx\\right)" }, { "type": "interim", "title": "Simplify $$2\\left(1\\cdot\\:e^{x^{2}}+e^{x^{2}}\\cdot\\:2xx\\right):{\\quad}2\\left(e^{x^{2}}+2e^{x^{2}}x^{2}\\right)$$", "input": "2\\left(1\\cdot\\:e^{x^{2}}+e^{x^{2}}\\cdot\\:2xx\\right)", "result": "=2\\left(e^{x^{2}}+2e^{x^{2}}x^{2}\\right)", "steps": [ { "type": "interim", "title": "$$1\\cdot\\:e^{x^{2}}=e^{x^{2}}$$", "input": "1\\cdot\\:e^{x^{2}}", "steps": [ { "type": "step", "primary": "Multiply: $$1\\cdot\\:e^{x^{2}}=e^{x^{2}}$$", "result": "=e^{x^{2}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7X+dJ/U92NOLIkjWa+WoQiVTiRdarS5WaUwjWcvuHZnijkVi15I8rBefLi4Iyt2wrUK+2oeUPEcqa43NTnHUMqjya2IcHPs77h6zi8+fDJdpar4B+sFqKWwU+IPpJd6JmWJUje683VvL4A301+OgoVQ==" } }, { "type": "interim", "title": "$$e^{x^{2}}\\cdot\\:2xx=2e^{x^{2}}x^{2}$$", "input": "e^{x^{2}}\\cdot\\:2xx", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$xx=\\:x^{1+1}$$" ], "result": "=e^{x^{2}}\\cdot\\:2x^{1+1}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=e^{x^{2}}\\cdot\\:2x^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7WKiT6jMjb052tPAUj4O+q0HLcpp2NIY1JK6NIX9t1+pwkKGJWEPFPk38sdJMsyPIOLamAczAscrQW/lgSM019ODYIwvGcMIl55QpEg6m8Y/wRIuvPR7Uys8rFryqi3z9vm6FlCuynYu9RsKay9FuRcLFnHKiAFvgIgHTqXwmbIw=" } }, { "type": "step", "result": "=2\\left(2e^{x^{2}}x^{2}+e^{x^{2}}\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7L0BOnEdP34gvl5LMjc8ilD+JpJ9t0c1RUZcBEdMBCjxdJAcqPp39MLcH0X5+QWhIzRqDxPUzBN6vjj5oJL9kUGh3OeBV/vtasWwIlK/J+Ph0xg/tdo2Rs5J5eb81eHodZuSdDtSSnN9l7lJRRFVUnu5AIz++qluupTlLFEcE9J3mn1CDLGvKU/ON0mVvnnalHzxcKnvcW0U3AaL7CtTDbndUTsEOMLquk4u3sKTBeFgkt3WiGR7ZaCaXvz77bMjS" } } ], "meta": { "solvingClass": "Derivatives", "practiceLink": "/practice/derivatives-practice", "practiceTopic": "Derivatives" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "y=2(e^{x^{2}}+2e^{x^{2}}x^{2})" }, "showViewLarger": true } }, "meta": { "showVerify": true } }