derivative of-1/(1+x^2)

解

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

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

解答ステップ

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

定数を除去:

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

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

指数の規則を適用する:

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

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

連鎖法則を適用:

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

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

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

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

簡素化

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

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