derivative of f(x)=x-3x^{1/3}

{ "query": { "display": "derivative of $$f\\left(x\\right)=x-3x^{\\frac{1}{3}}$$", "symbolab_question": "PRE_CALC#derivative f(x)=x-3x^{\\frac{1}{3}}" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Derivatives", "subTopic": "Derivatives", "default": "1-\\frac{1}{x^{\\frac{2}{3}}}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{d}{dx}\\left(x-3x^{\\frac{1}{3}}\\right)=1-\\frac{1}{x^{\\frac{2}{3}}}$$", "input": "\\frac{d}{dx}\\left(x-3x^{\\frac{1}{3}}\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{dx}{dx}-\\frac{d}{dx}\\left(3x^{\\frac{1}{3}}\\right)" }, { "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": "$$\\frac{d}{dx}\\left(3x^{\\frac{1}{3}}\\right)=\\frac{1}{x^{\\frac{2}{3}}}$$", "input": "\\frac{d}{dx}\\left(3x^{\\frac{1}{3}}\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=3\\frac{d}{dx}\\left(x^{\\frac{1}{3}}\\right)" }, { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=3\\cdot\\:\\frac{1}{3}x^{\\frac{1}{3}-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "interim", "title": "Simplify $$3\\cdot\\:\\frac{1}{3}x^{\\frac{1}{3}-1}:{\\quad}\\frac{1}{x^{\\frac{2}{3}}}$$", "input": "3\\cdot\\:\\frac{1}{3}x^{\\frac{1}{3}-1}", "result": "=\\frac{1}{x^{\\frac{2}{3}}}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:3x^{\\frac{1}{3}-1}}{3}" }, { "type": "step", "primary": "Cancel the common factor: $$3$$", "result": "=1\\cdot\\:x^{\\frac{1}{3}-1}" }, { "type": "interim", "title": "$$x^{\\frac{1}{3}-1}=x^{-\\frac{2}{3}}$$", "input": "x^{\\frac{1}{3}-1}", "steps": [ { "type": "interim", "title": "Join $$\\frac{1}{3}-1:{\\quad}-\\frac{2}{3}$$", "input": "\\frac{1}{3}-1", "result": "=x^{-\\frac{2}{3}}", "steps": [ { "type": "step", "primary": "Convert element to fraction: $$1=\\frac{1\\cdot\\:3}{3}$$", "result": "=-\\frac{1\\cdot\\:3}{3}+\\frac{1}{3}" }, { "type": "step", "primary": "Since the denominators are equal, combine the fractions: $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{-1\\cdot\\:3+1}{3}" }, { "type": "interim", "title": "$$-1\\cdot\\:3+1=-2$$", "input": "-1\\cdot\\:3+1", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:3=3$$", "result": "=-3+1" }, { "type": "step", "primary": "Add/Subtract the numbers: $$-3+1=-2$$", "result": "=-2" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7iSe/oPZ15XQklhwmxDEIoVXTSum/z5kLpMzXS1UJIexttTTHAByi5+zcJ9JCN6ZAyCE30rzMlUAIVDyhseMBrufcnznbZfVbgKLGaldkjXU=" } }, { "type": "step", "result": "=\\frac{-2}{3}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{2}{3}" } ], "meta": { "interimType": "Algebraic Manipulation Join Concise Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7DborvwU589/20W5rwUAp2+0se7vRyav6BwUCJZptwG3MwViaLUXkeD+JukROhWdjeVW9/X7jmVoO3IMaXztB2wH2kDe5DGYTz3TrPquGdIhR/IWXCqoVxn0KSq5yq0Z+6M8osviUPEkWv33aMbZrSIBcZlU2JZxWsEokVU5V0NU=" } }, { "type": "step", "result": "=1\\cdot\\:x^{-\\frac{2}{3}}" }, { "type": "step", "primary": "Apply exponent rule: $$a^{-b}=\\frac{1}{a^b}$$", "secondary": [ "$$x^{-\\frac{2}{3}}=\\frac{1}{x^{\\frac{2}{3}}}$$" ], "result": "=1\\cdot\\:\\frac{1}{x^{\\frac{2}{3}}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:\\frac{1}{x^{\\frac{2}{3}}}=\\frac{1}{x^{\\frac{2}{3}}}$$", "result": "=\\frac{1}{x^{\\frac{2}{3}}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7CzXS+MvaOUdKL6IkXGMFAxuAhzfH7ZbhM8JXdppsXF3jt0YWJlZUdkIYOP42z9IldYPfXQvX4/bINBB8wSEQ0U1Y7SwT225b4rPxVIUQTzgnA6BN4Zi/43DUpsJDK6CbP8vQyhiD4JSfqjIvcQ7timkSOxgqdB0M/sw8Nt2sXXRazroBYpdJ8ysZOU8vZ64sQPzAZEvjTNZd6Dsb74kOlDGnaEkdXzmzeDi9NT6Lpkg=" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=1-\\frac{1}{x^{\\frac{2}{3}}}" } ], "meta": { "solvingClass": "Derivatives", "practiceLink": "/practice/derivatives-practice", "practiceTopic": "Derivatives" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "y=1-\\frac{1}{x^{\\frac{2}{3}}}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }