derivative of 2/(\sqrt[3]{x-1})

解

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

- \frac{2}{3 ( x - 1 ) ^{\frac{4}{3}}}

解答ステップ

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

定数を除去:

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

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

指数の規則を適用する:

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

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

連鎖法則を適用:

- \frac{1}{3 ( x - 1 ) ^{\frac{4}{3}}} \frac{d}{d x} (x - 1)

= - \frac{1}{3 ( x - 1 ) ^{\frac{4}{3}}} \frac{d}{d x} (x - 1)

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

= 2 (- \frac{1}{3 ( x - 1 ) ^{\frac{4}{3}}} \cdot 1)

簡素化

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

= - \frac{2}{3 ( x - 1 ) ^{\frac{4}{3}}}