derivative of (e^x/((1+e^{2x))^{3/2}})

解

\frac{d}{d x} (\frac{e ^{x}}{( 1 + e ^{2 x} ) ^{\frac{3}{2}}})

\frac{e ^{x} ( - 2 e ^{2 x} + 1 )}{( 1 + e ^{2 x} ) ^{\frac{5}{2}}}

解答ステップ

\frac{d}{d x} (\frac{e ^{x}}{( 1 + e ^{2 x} ) ^{\frac{3}{2}}})

商の法則を適用:

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

= \frac{\frac{d}{d x} ( e ^{x} ) ( 1 + e ^{2 x} ) ^{\frac{3}{2}} - \frac{d}{d x} ( ( 1 + e ^{2 x} ) ^{\frac{3}{2}} ) e ^{x}}{( ( 1 + e ^{2 x} ) ^{\frac{3}{2}} ) ^{2}}

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

\frac{d}{d x} ((1 + e^{2 x})^{\frac{3}{2}}) = 3 e^{2 x} (1 + e^{2 x})^{\frac{1}{2}}

= \frac{e ^{x} ( 1 + e ^{2 x} ) ^{\frac{3}{2}} - 3 e ^{2 x} ( 1 + e ^{2 x} ) ^{\frac{1}{2}} e ^{x}}{( ( 1 + e ^{2 x} ) ^{\frac{3}{2}} ) ^{2}}

簡素化

\frac{e ^{x} ( 1 + e ^{2 x} ) ^{\frac{3}{2}} - 3 e ^{2 x} ( 1 + e ^{2 x} ) ^{\frac{1}{2}} e ^{x}}{( ( 1 + e ^{2 x} ) ^{\frac{3}{2}} ) ^{2}} : \frac{e ^{x} ( - 2 e ^{2 x} + 1 )}{( 1 + e ^{2 x} ) ^{\frac{5}{2}}}

= \frac{e ^{x} ( - 2 e ^{2 x} + 1 )}{( 1 + e ^{2 x} ) ^{\frac{5}{2}}}