I have exactly the same question. The equation is even more or less the same. Mine is:
f(x,y)=\frac{xy(x^2-y^2)}{x^2+y^2} if (x,y) \neq (0,0)
f(x,y)=0 if (x,y)=(0,0).
But I'm still puzzled how to threat the point (0,0). According to my textbook, f_{xy}(0,0)\neq f_{yx}(0,0). So, I know...