derivative of (e^{1/x}/x)

解

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

\frac{- e ^{\frac{1}{x}} - e ^{\frac{1}{x}} x}{x ^{3}}

解答ステップ

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

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

\frac{d x}{d x} = 1

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

簡素化

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

= \frac{- e ^{\frac{1}{x}} - e ^{\frac{1}{x}} x}{x ^{3}}