对复合函数做偏导运算的时候一定要在最终结果中替换掉所有中间函数的符号

一、题目

已知 $z=u^{2} \cos v$, $u=x y$, $v=2 x+y$, 则 $\frac{\partial z}{\partial x}=?$, $\frac{\partial z}{\partial y}=?$

难度评级：

二、解析

(1):

$$
\frac{\partial z}{\partial x}=\frac{\partial z}{\partial u} \cdot \frac{\partial u}{\partial x}+\frac{\partial z}{\partial v} \cdot \frac{\partial v}{\partial x}=
$$

$$
2 u \cdot \cos v \cdot y+u^{2}(-\sin v) \cdot 2 =
$$

$$
\textcolor{orange}{
2 x y^{2} \cos (2 x+y)-2 x^{2} y^{2} \sin (2 x+y)
}
$$

(2):

$$
\frac{\partial z}{\partial y}=\frac{\partial z}{\partial u} \cdot \frac{\partial u}{\partial y}+\frac{\partial z}{\partial v} \cdot \frac{\partial v}{\partial x}=
$$

$$
2 u \cdot \cos v \cdot x+u^{2}(-\sin v) \cdot 1=
$$

$$
\textcolor{orange}{
2 x^{2} y \cos (2 x+y) x^{2} y^{2} \sin (2 x+y)
}
$$

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