\[ f(x,y)=\frac{-(x^{2}-1)(y^{2}-4)-x^{2}-y^{2}+5}{(x^{2}+y^{2}+1)^{2}} \]

\[ \partial x = \frac{2x(x^{2}(y^2-3)-y^4+2y^2+1)}{(x^2+y^2+1)^3} \]

\[ \partial y = -\frac{2y(x^4-x^2(y^2-7)+2)}{(x^2+y^2+1)^3} \]