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Homework Help: Parcial derivation of two variable function

  1. Oct 22, 2012 #1
    1. The problem statement, all variables and given/known data

    Given the function f(x) defined as:

    (x^3-y^3)/(x^2+y^2) if (x,y)≠(0,0)
    0 if (x,y)=(0,0)

    Find the parcial derivatives of the function at the point (0,0).
    Is the function f differentiable?

    2. Relevant equations

    3. The attempt at a solution

    d/dx [ (x^3-y^3)/(x^2+y^2)] = [3x^2(x^2+y^2) - 2x(x^3-y^3)] / (x^2+y^2)^2 =
    = [x^4+3x^2*y^2+2xy^3]/(x^4+2x^2y^2+y^4)

    I can't find the way out of this indetermination... As to the second question, if the parcial derivatives exist then the function is differentiable in the point (0,0).
    I know from the solutions that the result will be 1 and -1.

    If anyone could point me in the right direction I'd really appreciate!

  2. jcsd
  3. Oct 22, 2012 #2


    User Avatar
    Science Advisor
    Homework Helper

    Hi Jalo! :smile:

    (try using the X2 button just above the Reply box :wink:)

    ∂/∂x means the derivative keeping y fixed

    so ∂/∂x at (0,0), or at (anything,0) is the derivative keeping y = 0 :wink:

    (and ∂/∂y at (0,0) is the derivative keeping x = 0 )

    sooo … ? :smile:
  4. Oct 22, 2012 #3
    Oh lol... I can't believe I didn't saw that!
    I was thinking as if it both x and y were tending to 0... I guess I'm spending too much time solving limits!

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