derivative of 1/(2+x^2)

解

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

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

解答ステップ

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

指数の規則を適用する:

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

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

連鎖法則を適用:

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

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

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

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

簡素化

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

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