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

解

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

\frac{6 e ^{\frac{1}{x}}}{x ^{2} ( 3 + e ^{\frac{1}{x}} ) ^{2}}

解答ステップ

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

商の法則を適用:

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

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

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

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

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

簡素化

\frac{\frac{e ^{\frac{1}{x}}}{x ^{2}} ( 3 + e ^{\frac{1}{x}} ) - ( - \frac{e ^{\frac{1}{x}}}{x ^{2}} ) ( 3 - e ^{\frac{1}{x}} )}{( 3 + e ^{\frac{1}{x}} ) ^{2}} : \frac{6 e ^{\frac{1}{x}}}{x ^{2} ( 3 + e ^{\frac{1}{x}} ) ^{2}}

= \frac{6 e ^{\frac{1}{x}}}{x ^{2} ( 3 + e ^{\frac{1}{x}} ) ^{2}}