Answer
$$\frac{\partial z}{\partial x}=-y\sin xy$$
$$\frac{\partial z}{\partial y}=-x\sin xy$$
Work Step by Step
We will use chain rule to solve both partial derivatives.
The partial derivative with respect to $x$ is:
$$\frac{\partial z}{\partial x}=\frac{\partial }{\partial x}(\cos xy)=-\sin xy\frac{\partial }{\partial x}(xy)=-y\sin xy$$
because $y$ is treated as a constant.
The partial derivative with respcet to $y$ is:
$$\frac{\partial z}{\partial y}=\frac{\partial }{\partial y}(\cos xy)=-\sin xy\frac{\partial }{\partial y}(xy)=-x\sin xy$$
because now $x$ is treated as a constant.
