(\partial)/(\partial x)(ue^{2x})

{ "query": { "display": "$$\\frac{\\partial\\:}{\\partial\\:x}\\left(ue^{2x}\\right)$$", "symbolab_question": "DERIVATIVE#\\frac{\\partial }{\\partial x}(ue^{2x})" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Derivatives", "subTopic": "Partial Derivatives", "default": "\\frac{\\partial }{\\partial x}(u)e^{2x}+e^{2x}\\cdot 2u", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:x}\\left(ue^{2x}\\right)=\\frac{\\partial\\:}{\\partial\\:x}\\left(u\\right)e^{2x}+e^{2x}\\cdot\\:2u$$", "input": "\\frac{\\partial\\:}{\\partial\\:x}\\left(ue^{2x}\\right)", "steps": [ { "type": "step", "primary": "Apply the Product Rule: $$\\left(f{\\cdot}g\\right)'=f'{\\cdot}g+f{\\cdot}g'$$", "secondary": [ "$$f=u,\\:g=e^{2x}$$" ], "result": "=\\frac{\\partial\\:}{\\partial\\:x}\\left(u\\right)e^{2x}+\\frac{\\partial\\:}{\\partial\\:x}\\left(e^{2x}\\right)u", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Product%20Rule", "practiceTopic": "Product Rule" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:x}\\left(e^{2x}\\right)=e^{2x}\\cdot\\:2$$", "input": "\\frac{\\partial\\:}{\\partial\\:x}\\left(e^{2x}\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}e^{2x}\\frac{\\partial\\:}{\\partial\\:x}\\left(2x\\right)$$", "input": "\\frac{\\partial\\:}{\\partial\\:x}\\left(e^{2x}\\right)", "result": "=e^{2x}\\frac{\\partial\\:}{\\partial\\:x}\\left(2x\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=e^{v},\\:\\:v=2x$$" ], "result": "=\\frac{\\partial\\:}{\\partial\\:v}\\left(e^{v}\\right)\\frac{\\partial\\:}{\\partial\\:x}\\left(2x\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:v}\\left(e^{v}\\right)=e^{v}$$", "input": "\\frac{\\partial\\:}{\\partial\\:v}\\left(e^{v}\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{\\partial\\:}{\\partial\\:v}\\left(e^{v}\\right)=e^{v}$$", "result": "=e^{v}" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAm78XQdiccdYNZFUr3qlNRHHI5S0StY1FdtOqqOPr0Te9fpuTWGA/bcx8V9AIGMLhjgq4Lxd6wfEqqn7isBB4lorzea8M9Hp5QtXq9EUDk+3rAS+CcUEgTtci3JLKvUQTBmRFYEhwKyvVrS/hQ+v39LJLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=e^{v}\\frac{\\partial\\:}{\\partial\\:x}\\left(2x\\right)" }, { "type": "step", "primary": "Substitute back $$v=2x$$", "result": "=e^{2x}\\frac{\\partial\\:}{\\partial\\:x}\\left(2x\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAkUQcsJHLWU6j3MiiASfoRu/M3hESAhJ79Q7OQNfwXSMNgFtoYbJnzv+zNETDbCQXMYYCqvl+e94pW032CAXDQMNZFvGc7UIqQxoEt41Cm4Ep3G/GREkml78tgv19XBlf32Cck6ykF/R4sTjfi2uxUgo3oe/oyhMy2+1TQhDBd2fzGrMGuaCGoCkBTn65ypufhm2/z8V92sLreSsUccmX/A" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:x}\\left(2x\\right)=2$$", "input": "\\frac{\\partial\\:}{\\partial\\:x}\\left(2x\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=2\\frac{\\partial\\:}{\\partial\\:x}\\left(x\\right)" }, { "type": "step", "primary": "Apply the common derivative: $$\\frac{\\partial\\:}{\\partial\\:x}\\left(x\\right)=1$$", "result": "=2\\cdot\\:1" }, { "type": "step", "primary": "Simplify", "result": "=2", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAno4QJ4HBk7oek4Ww/3d//tnFjOV6V4e2DrBKqW1EhFu2EtmEvdU5EIfaZixikaOU0OG38IleojCyebAtZy+3Tm9JiwEB0ZXmaMqMWNNpbCrpbGGfud1pShayFW5kHjQPuwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=e^{2x}\\cdot\\:2" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{\\partial\\:}{\\partial\\:x}\\left(u\\right)e^{2x}+e^{2x}\\cdot\\:2u" } ], "meta": { "solvingClass": "Derivatives", "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Partial%20Derivatives", "practiceTopic": "Partial Derivatives" } }, "meta": { "showVerify": true } }