derivative y= 1/(sqrt(x^2+1))

解

導関数

y = \frac{1}{x ^{2} + 1}

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

解答ステップ

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

指数の規則を適用する:

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

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

連鎖法則を適用:

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

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

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

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

簡素化

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

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