Answer
$$\frac{\partial f(x,y)}{\partial x}=2x$$
$$\frac{\partial f(x,y)}{\partial y}=-4y$$
Work Step by Step
The partial derivative with respect to $x$ is:
$$\frac{\partial f(x,y)}{\partial x}=\frac{\partial}{\partial x}(x^2-2y^2+4)=2x$$
because $y$ is treated as a constant.
The partial derivative with respect to $y$ is:
$$\frac{\partial f(x,y)}{\partial y}=\frac{\partial}{\partial y}(x^2-2y^2+4)=-4y$$
because now $x$ is treated as a constant.