(\partial)/(\partial x)(x^2e^{-x/y})

解答

\frac{\partial}{\partial x} (x^{2} e^{- \frac{x}{y}})

2 e^{- \frac{x}{y}} x - \frac{e ^{- \frac{x}{y}} x ^{2}}{y}

求解步骤

\frac{\partial}{\partial x} (x^{2} e^{- \frac{x}{y}})

将

y

视为常数

使用微分乘法定则:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = x^{2}, g = e^{- \frac{x}{y}}

= \frac{\partial}{\partial x} (x^{2}) e^{- \frac{x}{y}} + \frac{\partial}{\partial x} (e^{- \frac{x}{y}}) x^{2}

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

\frac{\partial}{\partial x} (e^{- \frac{x}{y}}) = - \frac{e ^{- \frac{x}{y}}}{y}

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

化简

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

= 2 e^{- \frac{x}{y}} x - \frac{e ^{- \frac{x}{y}} x ^{2}}{y}

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