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Find ∂/∂y ∂F/dx (0,0) given F(x,y)

  1. Oct 30, 2011 #1

    s3a

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    1. The problem statement, all variables and given/known data
    Attached.


    2. Relevant equations
    Just taking derivatives.


    3. 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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  2. jcsd
  3. Oct 31, 2011 #2
    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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