derivative 1/(sqrt(16+x^2))

解

導関数

\frac{1}{16 + x ^{2}}

- \frac{x}{( 16 + x ^{2} ) ^{\frac{3}{2}}}

解答ステップ

\frac{d}{d x} (\frac{1}{16 + x ^{2}})

指数の規則を適用する:

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

= \frac{d}{d x} ((16 + x^{2})^{- \frac{1}{2}})

連鎖法則を適用:

- \frac{1}{2 ( 16 + x ^{2} ) ^{\frac{3}{2}}} \frac{d}{d x} (16 + x^{2})

= - \frac{1}{2 ( 16 + x ^{2} ) ^{\frac{3}{2}}} \frac{d}{d x} (16 + x^{2})

\frac{d}{d x} (16 + x^{2}) = 2 x

= - \frac{1}{2 ( 16 + x ^{2} ) ^{\frac{3}{2}}} \cdot 2 x

簡素化

- \frac{1}{2 ( 16 + x ^{2} ) ^{\frac{3}{2}}} \cdot 2 x : - \frac{x}{( 16 + x ^{2} ) ^{\frac{3}{2}}}

= - \frac{x}{( 16 + x ^{2} ) ^{\frac{3}{2}}}