Find ∂/∂y ∂F/dx (0,0) given F(x,y)

[s3a]

Homework Statement

Attached.

Homework Equations

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.

 

[susskind_leon]

Try
d/dx (x*y*(8*x^2+4*y^2)/(x^2+y^2)) for x=0
and you get 4y
Next try
d/dy (x*y*(8*x^2+4*y^2)/(x^2+y^2)) for y=0
and you get 8x
So the second derivatives will give you different results.