derivative of (2-5x/(3x^{1/3)})

解答

\frac{d}{d x} (\frac{2 - 5 x}{3 x ^{\frac{1}{3}}})

\frac{- 10 x - 2}{9 x ^{\frac{4}{3}}}

求解步骤

\frac{d}{d x} (\frac{2 - 5 x}{3 x ^{\frac{1}{3}}})

将常数提出:

(a \cdot f)^{'} = a \cdot f^{'}

= \frac{1}{3} \frac{d}{d x} (\frac{2 - 5 x}{x ^{\frac{1}{3}}})

使用微分除法定则:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= \frac{1}{3} \cdot \frac{\frac{d}{d x} ( 2 - 5 x ) x ^{\frac{1}{3}} - \frac{d}{d x} ( x ^{\frac{1}{3}} ) ( 2 - 5 x )}{( x ^{\frac{1}{3}} ) ^{2}}

\frac{d}{d x} (2 - 5 x) = - 5

\frac{d}{d x} (x^{\frac{1}{3}}) = \frac{1}{3 x ^{\frac{2}{3}}}

= \frac{1}{3} \cdot \frac{( - 5 ) x ^{\frac{1}{3}} - \frac{1}{3 x ^{\frac{2}{3}}} ( 2 - 5 x )}{( x ^{\frac{1}{3}} ) ^{2}}

化简

\frac{1}{3} \cdot \frac{( - 5 ) x ^{\frac{1}{3}} - \frac{1}{3 x ^{\frac{2}{3}}} ( 2 - 5 x )}{( x ^{\frac{1}{3}} ) ^{2}} : \frac{- 10 x - 2}{9 x ^{\frac{4}{3}}}

= \frac{- 10 x - 2}{9 x ^{\frac{4}{3}}}

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