(\partial)/(\partial x)((e^x)/(1+e^x))

解

\frac{\partial}{\partial x} (\frac{e ^{x}}{1 + e ^{x}})

\frac{e ^{x}}{( 1 + e ^{x} ) ^{2}}

解答ステップ

\frac{\partial}{\partial x} (\frac{e ^{x}}{1 + e ^{x}})

商の法則を適用:

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

= \frac{\frac{\partial}{\partial x} ( e ^{x} ) ( 1 + e ^{x} ) - \frac{\partial}{\partial x} ( 1 + e ^{x} ) e ^{x}}{( 1 + e ^{x} ) ^{2}}

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

\frac{\partial}{\partial x} (1 + e^{x}) = e^{x}

= \frac{e ^{x} ( 1 + e ^{x} ) - e ^{x} e ^{x}}{( 1 + e ^{x} ) ^{2}}

簡素化

\frac{e ^{x} ( 1 + e ^{x} ) - e ^{x} e ^{x}}{( 1 + e ^{x} ) ^{2}} : \frac{e ^{x}}{( 1 + e ^{x} ) ^{2}}

= \frac{e ^{x}}{( 1 + e ^{x} ) ^{2}}