derivative of (40)/(x^9)

{ "query": { "display": "derivative of $$\\frac{40}{x^{9}}$$", "symbolab_question": "PRE_CALC#derivative \\frac{40}{x^{9}}" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Derivatives", "subTopic": "Derivatives", "default": "-\\frac{360}{x^{10}}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{d}{dx}\\left(\\frac{40}{x^{9}}\\right)=-\\frac{360}{x^{10}}$$", "input": "\\frac{d}{dx}\\left(\\frac{40}{x^{9}}\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=40\\frac{d}{dx}\\left(\\frac{1}{x^{9}}\\right)" }, { "type": "step", "primary": "Apply exponent rule: $$\\frac{1}{a}=a^{-1}$$", "result": "=40\\frac{d}{dx}\\left(x^{-9}\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=40\\left(-9x^{-9-1}\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "interim", "title": "Simplify $$40\\left(-9x^{-9-1}\\right):{\\quad}-\\frac{360}{x^{10}}$$", "input": "40\\left(-9x^{-9-1}\\right)", "result": "=-\\frac{360}{x^{10}}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=-40\\cdot\\:9x^{-9-1}" }, { "type": "step", "primary": "Subtract the numbers: $$-9-1=-10$$", "result": "=-40\\cdot\\:9x^{-10}" }, { "type": "step", "primary": "Apply exponent rule: $$a^{-b}=\\frac{1}{a^b}$$", "secondary": [ "$$x^{-10}=\\frac{1}{x^{10}}$$" ], "result": "=40\\cdot\\:9\\cdot\\:\\frac{1}{x^{10}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=-\\frac{1\\cdot\\:40\\cdot\\:9}{x^{10}}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:40\\cdot\\:9=360$$", "result": "=-\\frac{360}{x^{10}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ErXXeakVnO6tb6OFuQdvRgCWKUbvV6WK3fDUgFtg3Q+Vu0y/nymFFvapFvN0D5+87Wc51CFHsGe4k3MVakm2ov8//6/nV5O4fb8Xgwi7mapyhd7tjiG+GxQNxDvGkZUlz8IBMlCrS0aHrYwVRJH4H7++BkIi2qZNiO8mRKzoqi8=" } } ], "meta": { "solvingClass": "Derivatives", "practiceLink": "/practice/derivatives-practice", "practiceTopic": "Derivatives" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "y=-\\frac{360}{x^{10}}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }