Answer
$$\frac{\partial z}{\partial x}=y^2\sinh xy^2$$
$$\frac{\partial z}{\partial y}=2xy\sinh xy^2$$
Work Step by Step
We will use chain rule to find both partial derivatives.
The partial derivative with respect to $x$ is:
$$\frac{\partial z}{\partial x}=\frac{\partial }{\partial x}(\cosh xy^2)=\sinh xy^2\frac{\partial }{\partial x}(xy^2)=\sinh xy^2\cdot y^2=y^2\sinh xy^2$$
because $y$ is treated as a constant.
The partial derivative with respect to $y$ is:
$$\frac{\partial z}{\partial y}=\frac{\partial }{\partial y}(\cosh xy^2)=\sinh xy^2\frac{\partial }{\partial y}(xy^2)=\sinh xy^2\cdot2xy=2xy\sinh xy^2$$
because now $x$ is treated as a constant.
