derivative ((e^{b/x}))/((e^{b/x)-1)}

解

導関数

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

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

解答ステップ

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

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

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

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

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

簡素化

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

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