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Derivatives of absolute values

  1. Sep 18, 2007 #1
    1. The problem statement, all variables and given/known data
    Where is the function f(x) = |x| differentiable?

    2. Relevant equations
    [f(x+h) - f(x)] / h

    3. The attempt at a solution
    I know that the graph of f(x)=|x| shows a corner at the origin from which 2 lines project at opposite slopes, as in they are symmetric about the y-axis.
    I've seen the solution but I don't understand why f(x) is differentiable for every number except 0.
  2. jcsd
  3. Sep 18, 2007 #2


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    Science Advisor
    Homework Helper

    |x|=x for x>0 and |x|=(-x) for x<0. x and -x are both differentiable. Remember the derivative expression is a limit. It's only correct in the limit where h->0. Think of h as REALLY SMALL.
  4. Sep 18, 2007 #3
    You can re-write |x| as a piecewise function

    f(x) = -x, x<0
    f(x) = 0, x=0
    f(x) = x, x>0

    Find the derivative and I think you'll see your answer -- Oh and you might want to read up on what a "cusp" is.
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