2012年考研数二第11题解析

题目

设 $z = f(\ln x + \frac{1}{y})$, 其中函数 $f(u)$ 可微,则 $x \frac{\partial z}{\partial x} + y^{2} \frac{\partial z}{\partial y} = ?$

解析

设 $u = \ln x + \frac{1}{y}$, 则:

$$
z = f(u).
$$

于是:

$$
\frac{\partial z}{\partial x} =
$$

$$
\frac{\partial f(u)}{\partial u} \frac{\partial u}{\partial x} =
$$

$$
\frac{\partial f(u)}{\partial u} \frac{1}{x};
$$

$$
\frac{\partial z}{\partial y} =
$$

$$
\frac{\partial f(u)}{\partial u} \frac{\partial u}{\partial y} =
$$

$$
\frac{\partial f(u)}{\partial u} (-\frac{1}{y^{2}}).
$$

于是:

$$
x \frac{\partial z}{\partial x} + y^{2} \frac{\partial z}{\partial y} =
$$

$$
\frac{\partial f(u)}{\partial u} – \frac{\partial f(u)}{\partial u} = 0.
$$

综上可知,正确答案为 $0$.

EOF


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