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Partial Derivative Homework

  1. Oct 15, 2008 #1
    1. The problem statement, all variables and given/known data

    Show that the function is not differentiable at (0,0).

    f(x,y) = [ (xy)/(x2 + y2)(1/2) if (x,y) =/ (0,0)

    [ 0 if (x,y) = (0,0)

    3. The attempt at a solution

    I know that the partial derivatives at point (0,0) = 0, so I don't know why the function is not differentiable at (0,0). Is there a certain equation that will help me prove that?

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

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
  3. Oct 16, 2008 #2


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    You titled this "partial differential homework" but it is important to understand that has very little to do with partial derivatives. "Differentiable" is NOT a matter of having partial derivatives.

    I think it is really important that you look up the definition of "differentiable" for functions of two variables. In Calculus of one variable, we typically define the "derivative" as a limit and then say that a function is "differentiable" if and only if that limit exists. In Calculus of more than one variable, it is standard to define "differentiable" separately from just the partial derivatives.

    I know several equivalent definitions of "differentiable" for two variables but I don't know which one your textbook is using: look it up please.
  4. Oct 16, 2008 #3


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    Homework Helper

    Hi klopez! :smile:

    It's because existence of partial derivatives ∂f/∂x and ∂f/∂y only prove differentiablity in the x and y directions.

    Hint: try some other direction. :wink:
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