derivative of 1/(2sqrt(3x-1))

解

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

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

解答ステップ

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

定数を除去:

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

= \frac{1}{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}

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

連鎖法則を適用:

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

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

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

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

簡素化

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

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