Answer
$$\frac{\partial z}{\partial x}=\sqrt y$$
$$\frac{\partial z}{\partial y}=\frac{x}{2\sqrt y}$$
Work Step by Step
The partial derivative with the respect to $x$ is:
$$\frac{\partial z}{\partial x}=\frac{\partial }{\partial x}(x\sqrt y)=\sqrt y\frac{\partial }{\partial x}(x)=\sqrt y\cdot1=\sqrt y$$
because $y$ is treated as a constant.
The partial derivative with the respect to $y$ is:
$$\frac{\partial z}{\partial y}=\frac{\partial }{\partial y}(x\sqrt y)=x\frac{\partial }{\partial y}(\sqrt y)=x\frac{1}{2\sqrt y}$$
because now $x$ is treated as a constant.
