z=cos(2xy) 的二阶偏导数计算详解

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：

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

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

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