d/(d{x)}(({z})/(({x)^2+{y}^2+{z}^2)^{3/2}})

解

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

- \frac{3 z x}{( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{5}{2}}}

解答ステップ

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

定数を除去:

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

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

指数の規則を適用する:

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

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

連鎖法則を適用:

- \frac{3}{2 ( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{5}{2}}} \frac{d}{d x} (x^{2} + y^{2} + z^{2})

= - \frac{3}{2 ( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{5}{2}}} \frac{d}{d x} (x^{2} + y^{2} + z^{2})

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

= z (- \frac{3}{2 ( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{5}{2}}} \cdot 2 x)

簡素化

z (- \frac{3}{2 ( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{5}{2}}} \cdot 2 x) : - \frac{3 z x}{( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{5}{2}}}

= - \frac{3 z x}{( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{5}{2}}}