d/(d{x)}(({x}^2+{y}^2+{z}^2)^{(-1)/2})

解

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

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

解答ステップ

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

連鎖法則を適用:

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

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

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

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

簡素化

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

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