derivative v/(v+c/v)

解

導関数

\frac{v}{v + \frac{c}{v}}

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

解答ステップ

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

商の法則を適用:

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

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

\frac{d v}{d v} = 1

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

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

簡素化

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

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