Answer
$$\frac{\partial z}{\partial x}=2xe^{2y}$$
$$\frac{\partial z}{\partial y}=2x^2e^{2y}$$
Work Step by Step
The partial derivative with respect to $x$ is:
$$\frac{\partial z}{\partial x}=\frac{\partial }{\partial x}(x^2e^{2y})=e^{2y}\frac{\partial }{\partial x}(x^2)=2xe^{2y}$$
Here $y$ is treated as a constant, so $e^{2y}$ is treated as a constant as well.
The partial derivative with respect to $y$ is:
$$\frac{\partial z}{\partial y}=\frac{\partial }{\partial y}(x^2e^{2y})=x^2\frac{\partial }{\partial y}(e^{2y})=x^2e^{2y}\frac{\partial }{\partial y}(2y)=2x^2e^{2y}$$
Here $x$ is treated as a constant, so $x^2$ is treated as a constant as well. Also, we used the chain rule to find $\frac{\partial }{\partial y}(e^{2y}).$