d/(d{y)}(({y})/({y)+c/({y)}})

解

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

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

解答ステップ

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

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

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

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

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

簡素化

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

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