For
$$f(x,y) = \begin{cases}
\frac{xy}{x^2+y^2} &\text{if }(x,y)\neq (0,0);\\
0 &\text{if }(x,y)=(0,0).\end{cases}$$
I'm trying to prove that $\frac{\partial f}{\partial x}$, the partial derivative with respect to $x$, exists.
Taking the partial derivative freehand I get
$$\frac{y(x^2+y^2)-xy(2x)}{(x^2+y^2)^2}=
\frac{y(y^2-x^2)}{(x^2+y^2)^2}.$$
But using the limits I get $$\lim_{x\to c}\frac{f(x,y)-f(c,y)}{x-c} = y(y^2-c^2).$$
I'm not sure what I'm doing wrong...