derivative u/(u+c/u)

解

導関数

\frac{u}{u + \frac{c}{u}}

\frac{2 c u}{( u ^{2} + c ) ^{2}}

解答ステップ

\frac{d}{d u} (\frac{u}{u + \frac{c}{u}})

商の法則を適用:

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

= \frac{\frac{d u}{d u} ( u + \frac{c}{u} ) - \frac{d}{d u} ( u + \frac{c}{u} ) u}{( u + \frac{c}{u} ) ^{2}}

\frac{d u}{d u} = 1

\frac{d}{d u} (u + \frac{c}{u}) = 1 - \frac{c}{u ^{2}}

= \frac{1 \cdot ( u + \frac{c}{u} ) - ( 1 - \frac{c}{u ^{2}} ) u}{( u + \frac{c}{u} ) ^{2}}

簡素化

\frac{1 \cdot ( u + \frac{c}{u} ) - ( 1 - \frac{c}{u ^{2}} ) u}{( u + \frac{c}{u} ) ^{2}} : \frac{2 c u}{( u ^{2} + c ) ^{2}}

= \frac{2 c u}{( u ^{2} + c ) ^{2}}