Answer
$$\frac{\partial f(x,y)}{\partial x}=2xy^3$$
$$\frac{\partial f(x,y)}{\partial y}=3x^2y^2$$
Work Step by Step
The partial derivative with respect to $x$ is:
$$\frac{\partial f(x,y)}{\partial x}=\frac{\partial}{\partial x}(x^2y^3)=y^3\frac{\partial}{\partial x}(x^2)=2xy^3$$
because $y$ is treated as a constant.
The partial derivative with respect to $y$ is:
$$\frac{\partial f(x,y)}{\partial y}=\frac{\partial }{\partial y}(x^2y^3)=x^2\frac{\partial }{\partial y}(y^3)=3x^2y^2$$
because now $x$ is treated as a constant.