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

解

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

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

解答ステップ

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

指数の規則を適用する:

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

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

連鎖法則を適用:

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

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

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

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

簡素化

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

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