derivative of f(x)=(x^2+1)^2

{ "query": { "display": "derivative of $$f\\left(x\\right)=\\left(x^{2}+1\\right)^{2}$$", "symbolab_question": "PRE_CALC#derivative f(x)=(x^{2}+1)^{2}" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Derivatives", "subTopic": "Derivatives", "default": "4x(x^{2}+1)", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{d}{dx}\\left(\\left(x^{2}+1\\right)^{2}\\right)=4x\\left(x^{2}+1\\right)$$", "input": "\\frac{d}{dx}\\left(\\left(x^{2}+1\\right)^{2}\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}2\\left(x^{2}+1\\right)\\frac{d}{dx}\\left(x^{2}+1\\right)$$", "input": "\\frac{d}{dx}\\left(\\left(x^{2}+1\\right)^{2}\\right)", "result": "=2\\left(x^{2}+1\\right)\\frac{d}{dx}\\left(x^{2}+1\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{2},\\:\\:u=\\left(x^{2}+1\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{2}\\right)\\frac{d}{dx}\\left(\\left(x^{2}+1\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{2}\\right)=2u$$", "input": "\\frac{d}{du}\\left(u^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2u^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2u", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYr+VZUwBnLdzbS6DZQ+f4s+k3hxk9aCfAWodBRxXgUexMchyqTAJWrzJaDbnNcFsJUeCBKuYKgaNJ253gLI69U79qbCA2QqVmvm3jGRXZ2ppvbGT4j1utMEkCDH25m/vlQ==" } }, { "type": "step", "result": "=2u\\frac{d}{dx}\\left(\\left(x^{2}+1\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\left(x^{2}+1\\right)$$", "result": "=2\\left(x^{2}+1\\right)\\frac{d}{dx}\\left(x^{2}+1\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYkqB1UNa2an0Zy0zONZ3grdYBXoei3w9mkdVlxI2fN/BOiaLJuL5RxgumX0gNvT19bcYBvJrr0UVRZhze7mTrdq/aFmkoUAaK8lDGfNdbcWmgNWcp2TT4w6Xuaszk/TmlIcpVkhDeBRyHbkBEvLVwsvwt9LEn7QCBUukJKctfSJKzbNLkYZsJlrVfdrZPnP/Gbv5g0x3AM4gImHjTs45aCY=" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(x^{2}+1\\right)=2x$$", "input": "\\frac{d}{dx}\\left(x^{2}+1\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{d}{dx}\\left(x^{2}\\right)+\\frac{d}{dx}\\left(1\\right)" }, { "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": "$$\\frac{d}{dx}\\left(1\\right)=0$$", "input": "\\frac{d}{dx}\\left(1\\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+idP5vLVrjUll6eMdYmqQX14xoif/Hxcm4iYenIFJ8Vk6wvKjVnTtwWT18bQnz7FeFrf3rcM8IZlDz2c0dm5O2bEw0Ql6ne7k1AUriTsKfyXa6Zj1lcQsTYejuhcz" } }, { "type": "step", "result": "=2x+0" }, { "type": "step", "primary": "Simplify", "result": "=2x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=2\\left(x^{2}+1\\right)\\cdot\\:2x" }, { "type": "step", "primary": "Simplify", "result": "=4x\\left(x^{2}+1\\right)", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "practiceLink": "/practice/derivatives-practice", "practiceTopic": "Derivatives" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "y=4x(x^{2}+1)" }, "showViewLarger": true } }, "meta": { "showVerify": true } }