(\partial)/(\partial y)((e^y)/(e^x+e^y))

解

\frac{\partial}{\partial y} (\frac{e ^{y}}{e ^{x} + e ^{y}})

\frac{e ^{x + y}}{( e ^{x} + e ^{y} ) ^{2}}

解答ステップ

\frac{\partial}{\partial y} (\frac{e ^{y}}{e ^{x} + e ^{y}})

x

を定数として扱う

商の法則を適用:

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

= \frac{\frac{\partial}{\partial y} ( e ^{y} ) ( e ^{x} + e ^{y} ) - \frac{\partial}{\partial y} ( e ^{x} + e ^{y} ) e ^{y}}{( e ^{x} + e ^{y} ) ^{2}}

\frac{\partial}{\partial y} (e^{y}) = e^{y}

\frac{\partial}{\partial y} (e^{x} + e^{y}) = e^{y}

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

簡素化

\frac{e ^{y} ( e ^{x} + e ^{y} ) - e ^{y} e ^{y}}{( e ^{x} + e ^{y} ) ^{2}} : \frac{e ^{x + y}}{( e ^{x} + e ^{y} ) ^{2}}

= \frac{e ^{x + y}}{( e ^{x} + e ^{y} ) ^{2}}