求偏导时，函数的第一部分变量用 1 表示，第二部分变量用 2 表示

一、题目

已知 $z$ $=$ $\mathrm{e}^{x y}$ $+$ $f(x+y, x y)$, $f(u, v)$ 有二阶连续偏导数, 则 $\frac{\partial^2 z}{\partial x \partial y}$ $=$ $?$

难度评级：

二、解析

$$
\frac{\partial z}{\partial x} = y e^{xy} + f^{\prime}_{1} + y f^{\prime}_{2} \Rightarrow
$$

$$
\frac{\partial ^{2} z}{\partial x \partial y} = e^{x y} + xy e^{xy} + f^{\prime \prime}_{11} + xf^{\prime \prime}_{12} + f^{\prime}_{2} + yf^{\prime \prime}_{21} + xy f^{\prime \prime}_{22}
$$

Next

又：

$$
f^{\prime \prime}_{12} = f^{\prime \prime}_{21}
$$

于是：

$$
\frac{\partial ^{2} z}{\partial x \partial y} = e^{x y} + xy e^{xy} + f^{\prime \prime}_{11} + (x + y)f^{\prime \prime}_{12} + f^{\prime}_{2} + xy f^{\prime \prime}_{22}
$$

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