derivative of 9/(sqrt(x+8))

解

\frac{d}{d x} (\frac{9}{x + 8})

- \frac{9}{2 ( x + 8 ) ^{\frac{3}{2}}}

解答ステップ

\frac{d}{d x} (\frac{9}{x + 8})

定数を除去:

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

= 9 \frac{d}{d x} (\frac{1}{x + 8})

指数の規則を適用する:

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

= 9 \frac{d}{d x} ((x + 8)^{- \frac{1}{2}})

連鎖法則を適用:

- \frac{1}{2 ( x + 8 ) ^{\frac{3}{2}}} \frac{d}{d x} (x + 8)

= - \frac{1}{2 ( x + 8 ) ^{\frac{3}{2}}} \frac{d}{d x} (x + 8)

\frac{d}{d x} (x + 8) = 1

= 9 (- \frac{1}{2 ( x + 8 ) ^{\frac{3}{2}}} \cdot 1)

簡素化

9 (- \frac{1}{2 ( x + 8 ) ^{\frac{3}{2}}} \cdot 1) : - \frac{9}{2 ( x + 8 ) ^{\frac{3}{2}}}

= - \frac{9}{2 ( x + 8 ) ^{\frac{3}{2}}}