求解全微分：$z$ $=$ $e^{\frac{y}{x}}$

一、题目

求解下面这个函数的全微分：

$$
z = e^{\frac{y}{x}}
$$

难度评级：

二、解析

$$
\frac{\partial z}{\partial x} = e^{\frac{y}{x}} \times \frac{-y}{x^{2}} = \frac{-y}{x^{2}} e^{\frac{y}{x}}.
$$

$$
\frac{\partial z}{\partial y} = e^{\frac{y}{x}} \times \frac{1}{x} = \frac{1}{x} e^{\frac{y}{x}}.
$$

Next

于是：

$$
\mathrm{d} z = \frac{-y}{x^{2}} e^{\frac{y}{x}} \mathrm{d} x + \frac{1}{x} e^{\frac{y}{x}} \mathrm{d} y.
$$

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