Answer
$$f_x=\frac{\partial f}{\partial x}=4y^3,f_y=\frac{\partial f}{\partial y}=12xy^2.$$
Work Step by Step
To calculate the partial derivatives, treat all variables as constant except the variable we are deriving with respect to.
Since $f(x,y)=4xy^3$, then we have
$$f_x=\frac{\partial f}{\partial x}=4y^3,$$
$$f_y=\frac{\partial f}{\partial y}=12xy^2.$$
