d/(dz)(z/((z^2+R^2)^{3/2)})

解

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

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

解答ステップ

\frac{d}{d z} (\frac{z}{( z ^{2} + R ^{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 z}{d z} ( z ^{2} + R ^{2} ) ^{\frac{3}{2}} - \frac{d}{d z} ( ( z ^{2} + R ^{2} ) ^{\frac{3}{2}} ) z}{( ( z ^{2} + R ^{2} ) ^{\frac{3}{2}} ) ^{2}}

\frac{d z}{d z} = 1

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

= \frac{1 \cdot ( z ^{2} + R ^{2} ) ^{\frac{3}{2}} - 3 ( z + R \frac{d R}{d z} ) ( z ^{2} + R ^{2} ) ^{\frac{1}{2}} z}{( ( z ^{2} + R ^{2} ) ^{\frac{3}{2}} ) ^{2}}

簡素化

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

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