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

解答

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

- \frac{2}{9 ( x - 1 ) ^{\frac{5}{3}}}

求解步骤

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

将常数提出:

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

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

使用指数法则:

\frac{1}{a} = a^{- 1}

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

使用连锁律:

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

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

\frac{d}{d x} (x - 1) = 1

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

化简

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

= - \frac{2}{9 ( x - 1 ) ^{\frac{5}{3}}}

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