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Rigorously determining differentiability in multiple variables

  1. May 28, 2013 #1
    1. The problem statement, all variables and given/known data

    Determine if

    f(x,y) = ((x-y)4 +x3 +xy2)/(x2+y2)
    [f(x,y = 0 @ (0,0)]

    is differentiable at the origin.

    2. Relevant equations

    x = (0,0)

    3. The attempt at a solution

    A function is differentiable at x if f(xx) - f(x) = AΔx + |Δx|R(x)

    Where A are constant coefficients of the vector Δx, and R(Δx) → 0 as Δx → 0.

    A couple of questions.

    1) in order to solve this, should I set A = [∂f/∂x, ∂f/∂y] evaluated at (0,0)? My literature claims A just needs to be "some constant", not literally the partials evaluated at that point. A bit confused there.

    2) Just to double check, if the first step is determining ∂f/∂x at a given point, I am free to set y=0 BEFORE calculating the partial wrt x, right?
     
  2. jcsd
  3. May 28, 2013 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    From the point of view of defining "differentiable", "A" just has to be some fixed vector. But then one can show that it is, in fact, the partials evaluated at that point.

    Yes, in calculating the partial derivative, at a given point, you hold all the other variables constant.

    In fact, an example of a function which has partial derivatives at a given point but is NOT differentiable there is f(x,y)= 0 if xy= 0, 1 otherwise. To find the partial derivative with respect to x, at (0, 0), we set y= 0 so xy= 0 for all x: f(x,0)= 0 for all x so has derivative 0. If y were allowed to have non-zero value, xy= 0 only for x= 0 so f(x,0)= 0 if x=0, 1 otherwise and is not differentiable.
     
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