Just taking derivatives.
The Attempt at a Solution
Basically, at first I was thinking I can differentiate because the original function is continuous at (0,0) so I do and get: http://www.wolframalpha.com/input/?i=d/dy+(d/dx+(xy(8x^2+%2B+4y^2)/(x^2%2By^2))). Correct me if I am wrong but I was thinking I had to differentiate this twice partially as the Wolfram Alpha link shows and then plug in (0,0) to the answer that Wolfram Alpha gave but that is also not defined. I tried taking the limit with y = 0 and x = 0 respectively and found that the limit as (x,y) => (0,0) of the differentiated part does not exist. I then thought that wait, having F(0,0) = 0 be defined does not mean that it is continuous so I shouldn't be able to differentiate in the first place. I am now completely confused as to what I need to do.