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

解答

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

- \frac{- e ^{\frac{1}{x}} - 2 e ^{\frac{1}{x}} x}{x ^{4}}

求解步骤

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

将常数提出:

(a \cdot f)^{'} = a \cdot f^{'}

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

使用微分除法定则:

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

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

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

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

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

化简

- \frac{( - \frac{e ^{\frac{1}{x}}}{x ^{2}} ) x ^{2} - 2 x e ^{\frac{1}{x}}}{( x ^{2} ) ^{2}} : - \frac{- e ^{\frac{1}{x}} - 2 e ^{\frac{1}{x}} x}{x ^{4}}

= - \frac{- e ^{\frac{1}{x}} - 2 e ^{\frac{1}{x}} x}{x ^{4}}

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