derivative (x^2)/((e^x-1)^2)

解

導関数

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

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

解答ステップ

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

\frac{d}{d x} (x^{2}) = 2 x

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

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

簡素化

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

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