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Math Help - Derivative

  1. #1
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    Derivative

    For the function f (x)=x|x| , show that f ' (0) exists. What is the value?
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  2. #2
    Super Member General's Avatar
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    Do you know the limit definition of the derivative of the function at point ?
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  3. #3
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    nop
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  4. #4
    Senior Member eumyang's Avatar
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    Hmm... you should! The derivative of f at c is
    f'(c) = \lim\limits_{x \to c} \dfrac{f(x) - f(c)}{x - c}.

    The existence of this limit in this form requires that the one-sided limits
    f'(c) = \lim\limits_{x \to c-} \dfrac{f(x) - f(c)}{x - c} and f'(c) = \lim\limits_{x \to c+} \dfrac{f(x) - f(c)}{x - c} exist and are equal. I think you will have to evaluate both one-sided limits and show that they are equal in order to prove that f'(0) exists in your case.
    Last edited by eumyang; August 8th 2010 at 10:18 PM.
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