I got 1/2y^2x for fxx, but think I did something wrong so I stopped. Please find fxx, fyy, fxy. Thanks.
Just to get you started, $\displaystyle f_x(x,y) = \frac{2xy^2}{1+x^2y^2}$. To find $\displaystyle f_{xx}$, use the quotient rule to differentiate $\displaystyle f_x$.