Let $\displaystyle f: \mathbb {R}^2 \rightarrow \mathbb {R} $ be defined by $\displaystyle f(x,y)= \frac {xy(x^2-y^2)}{x^2+y^2} $ if $\displaystyle (x,y) \neq (0,0) $ and $\displaystyle f(0,0)=0$

Show that $\displaystyle \frac { \partial ^2 f }{ \partial x \partial y } $ and $\displaystyle \frac { \partial ^2 f }{ \partial y \partial x } $ exist at (0,0) but not equal.

Now, should I go ahead and take the partial derivative as how it looks then plug in (0,0)? Or should I approach this with the limit?

I just want to know if I'm going to start this one correctly. Thank you!