z=cos2xy的二阶偏导数

对于 $z=\cos(2xy)$，有：

$$\frac{\partial z}{\partial x}=-2y\sin(2xy)$$

$$\frac{\partial^2 z}{\partial x^2}=-4y^2\cos(2xy)$$

$$\frac{\partial z}{\partial y}=-2x\sin(2xy)$$

$$\frac{\partial^2 z}{\partial y^2}=-4x^2\cos(2xy)$$

$$\frac{\partial^2 z}{\partial x\partial y}=-2\sin(2xy)+4x^2y\sin(2xy)=-2\sin(2xy)+2xy\frac{\partial z}{\partial x}$$

因此，$z=\cos(2xy)$ 的二阶偏导数为：

$$\frac{\partial^2 z}{\partial x^2}=-4y^2\cos(2xy)$$

$$\frac{\partial^2 z}{\partial y^2}=-4x^2\cos(2xy)$$

$$\frac{\partial^2 z}{\partial x\partial y}=-2\sin(2xy)+2xy(-2y\sin(2xy))=-2\sin(2xy)-4x^2y^2\sin(2xy)$$