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

解

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

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

解答ステップ

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

定数を除去:

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

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

指数の規則を適用する:

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

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

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

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

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

簡素化

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

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