derivative of xe^{y/x}

解答

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

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

求解步骤

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

使用微分乘法定则:

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

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

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

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

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

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

化简

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

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

x \to 1 lim ((1 - x)^{2} + 2)

\frac{d}{d x} (\frac{2160}{x ^{2}})

x \to \infty lim (\frac{4}{0})

\int_{0}^{4} x^{2} (0.5 - 0.125 x) d x

\frac{d}{d x} (tan (6 x^{2} + 3))