(\partial)/(\partial x)(1/(x^2+1))

解

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

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

解答ステップ

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

指数の規則を適用する:

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

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

連鎖法則を適用:

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

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

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

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

簡素化

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

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