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

解

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

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

解答ステップ

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

商の法則を適用:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

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

\frac{d}{d y} (y) = 1

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

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

簡素化

\frac{1 \cdot ( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{3}{2}} - 3 y ( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{1}{2}} y}{( ( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{3}{2}} ) ^{2}} : \frac{- 2 y ^{2} + x ^{2} + z ^{2}}{( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{5}{2}}}

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