例 9.32求 $f(x, y) = x^2y^3 + 3y + x$ 的二阶偏导数。

解 我们在例 9.27 中已经求得

$$\frac{\partial f}{\partial x} = 2xy^3 + 1 \quad \text{and} \quad \frac{\partial f}{\partial y} = 3x^2y^2 + 3$$

再次求导，我们得到

$$\begin{aligned} \frac{\partial}{\partial x} \left( \frac{\partial f}{\partial x} \right) &= \frac{\partial^2 f}{\partial x^2} = 2y^3, \quad \frac{\partial}{\partial y} \left( \frac{\partial f}{\partial x} \right) = \frac{\partial^2 f}{\partial y \partial x} = 6xy^2 \ \frac{\partial}{\partial y} \left( \frac{\partial f}{\partial y} \right) &= \frac{\partial^2 f}{\partial y^2} = 6x^2y, \quad \frac{\partial}{\partial x} \left( \frac{\partial f}{\partial y} \right) = \frac{\partial^2 f}{\partial x \partial y} = 6xy^2 \end{aligned}$$

注意在本例中

$$\frac{\partial^2 f}{\partial x \partial y} = \frac{\partial^2 f}{\partial y \partial x}$$

在 MATLAB 中，可以通过对已求得的一阶偏导数再进行适当的求导来得到二阶偏导数。因此在 MATLAB 中， $f(x, y)$ 的二阶偏导数由下式给出

$$\begin{aligned} f_{xx} &= \text{diff}(f_x, x), \quad f_{xy} = \text{diff}(f_x, y), \quad f_{yy} = \text{diff}(f_y, y), \ f_{yx} &= \text{diff}(f_y, x) \end{aligned}$$

或者，也可以直接使用以下命令求得非混合偏导数

$$f_{xx} = \text{diff}(f, x, 2), \quad f_{yy} = \text{diff}(f, y, 2)$$

并且可以推广到更高阶的偏导数。

考虑例 9.32，MATLAB 命令

syms x y
f = x^2*y^3 + 3*y + x;
fx = diff(f, x); fy = diff(f, y); fxx = diff(fx, x) return
fxx = 2*y^3
fxy = diff(fx, y) returns fxy = 6*x*y^2
fyy = diff(fy, y) returns fyy = 6*x^2*y
fyx = diff(fy, x) returns fyx = 6*x*y^2