Find ∂/∂y ∂F/dx (0,0) given F(x,y)

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SUMMARY

The discussion focuses on finding the mixed partial derivative ∂/∂y ∂F/∂x at the point (0,0) for the function F(x,y) = xy(8x² + 4y²)/(x² + y²). Participants highlight that while the function is defined at (0,0), its continuity is questionable, leading to confusion about the differentiability at that point. Attempts to compute the derivatives using Wolfram Alpha reveal that the limits do not exist, indicating that the function may not be differentiable at (0,0). The conclusion emphasizes the need to verify continuity before applying differentiation rules.

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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. :cry:
 

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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.
 

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