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