2021高等数学《微积分》摸底测试与答案-Kmath科数- 中考、高考、考验数学题库

设 $z=f\left(x-y, x^2 y\right), f$ 具有连续的二阶偏导数, 求 $\frac{\partial^2 z}{\partial x \partial y}$.

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答案：

解$$
\begin{aligned}
&\frac{\partial z}{\partial x}=f_1^{\prime}+2 x y f_2^{\prime} \\
& \frac{\partial^2 z}{\partial x \partial y}=\left(f_1^{\prime}\right)_y^{\prime}+2 x y\left(f_2^{\prime}\right)_y^{\prime}+(2 x y)_y^{\prime} f_2^{\prime} \\
& =-f_{11}^{\prime \prime}+x^2 f_{12}^{\prime \prime}-2 x y f_{21}^{\prime \prime}+2 x^3 y f_{22}^{\prime \prime}+2 x f_2^{\prime} \\
&
\end{aligned}
$$

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