derivative of C(x)=-0.01x^2+7.2x-163

{ "query": { "display": "derivative of $$C\\left(x\\right)=-0.01x^{2}+7.2x-163$$", "symbolab_question": "PRE_CALC#derivative C(x)=-0.01x^{2}+7.2x-163" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Derivatives", "subTopic": "Derivatives", "default": "-0.02x+7.2", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{d}{dx}\\left(-0.01x^{2}+7.2x-163\\right)=-0.02x+7.2$$", "input": "\\frac{d}{dx}\\left(-0.01x^{2}+7.2x-163\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=-\\frac{d}{dx}\\left(0.01x^{2}\\right)+\\frac{d}{dx}\\left(7.2x\\right)-\\frac{d}{dx}\\left(163\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(0.01x^{2}\\right)=0.02x$$", "input": "\\frac{d}{dx}\\left(0.01x^{2}\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=0.01\\frac{d}{dx}\\left(x^{2}\\right)" }, { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=0.01\\cdot\\:2x^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=0.02x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYgOlHcnqYdIlvf9cHoubuGDCRJgoAMiTd6o06hbXd0s3BOWqhjf7C2X3qBWUcLRi6lgSY9hEwFDLp33WBJ5LvWNPXEpfcfbu02Nx1nf7/MLih114MVLjHzTUZySfjMgxoEfKTxLDYs0w9qXNJPzqlDY=" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(7.2x\\right)=7.2$$", "input": "\\frac{d}{dx}\\left(7.2x\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=7.2\\frac{dx}{dx}" }, { "type": "step", "primary": "Apply the common derivative: $$\\frac{dx}{dx}=1$$", "result": "=7.2\\cdot\\:1" }, { "type": "step", "primary": "Simplify", "result": "=7.2", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYjTobLYFyKoPaXNvlnnnbK2cWM5XpXh7YOsEqpbUSEW7MaT46588w+Smtdno5kfBEUeCBKuYKgaNJ253gLI69U4dd/kF/93IPWmcLQXdb74dYdpRIiA0/zfQFwCgSwXaKw==" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(163\\right)=0$$", "input": "\\frac{d}{dx}\\left(163\\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+idP5vLVrjUll6eMdYmia4ar5GE41JEO/67IMtzOXIQHgliMhSOSNsNni19Inf7LqB9CcyvYCWDsGseX09s1bIZxfodm3UsZcfZAZr4v+f5+B8iT9ULmcCNTGOvNKJLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=-0.02x+7.2-0" }, { "type": "step", "primary": "Simplify", "result": "=-0.02x+7.2", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "practiceLink": "/practice/derivatives-practice", "practiceTopic": "Derivatives" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "y=-0.02x+7.2" }, "showViewLarger": true } }, "meta": { "showVerify": true } }