$$f(x,y)=\left\{\begin{matrix} xy\frac{x^{2}-2y^{2}}{x^{2}+y^{2}} & (x,y)\neq (0,0)\\ 0 & (x,y)=(0,0) \end{matrix}\right. $$ Calculate $ f_{xy} (0,0), f_{yx}(0,0) $.
- I found $f_{x}$ and then for $f_{xy}(0,0) $ I took the limit: $$f_{xy}(0,0) =\lim_{y\rightarrow 0}\frac{f_{x}(0,y)-f_{x}(0,0)}{y-0}$$ For $ f_{x}(0,0) $ I also took the limit: $$f_{x}(0,0) =\lim_{x\rightarrow 0}\frac{f(x,0)-f(0,0)}{x-0}$$ I got that $f_{x}(0,0) = 0 $ and $f_{x}(0,y) = -2y$ , so $$f_{xy}(0,0) = -2$$
- I did similar things for $f_{yx}(0,0) $ and I got that $$f_{yx}(0,0) = 0 $$
Can I be correct?