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Math Help - derivative of |x|

  1. #1
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    derivative of |x|

    I'm asked to find the derivative of f(x)=x|x| at the point (0,0).

    Would I do this using the definition of absolute value? o_O
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  2. #2
    Senior Member Twig's Avatar
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    hi

    Uhm, is it meant to say  f(x) = x|x| , and not
     f(x) = |x| ?

     f(x) = |x| is not differentiable in x = 0.

    \lim_{h \rightarrow 0} \frac{f(x+h) - f(x)}{h} does not exist in x = 0.
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  3. #3
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    Quote Originally Posted by Twig View Post
    Uhm, is it meant to say  f(x) = x|x|

    yes, thats is the question. but i dont get what to do from there... :\
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  4. #4
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    I would start this problem by piecewise defining f(x)=(x)|x|, which I think is what you meant by the definition of the absolute value. Then take the derivatives of each piecewise defined function. Looking at the graph of f(x)=(x)|x| will reveal that the function has no "problems" at x=0, i.e. it is continuous and smooth in the neigborhood surrounding the point x=0.

    Hope this helps.

    (P.S. I'm not sure if this is correct but this is where I would start if I were doing the problem)
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  5. #5
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    Do you see that \begin{array}{lcl}<br />
{x > 0} & \Rightarrow & {x\left| x \right| = x^2 } \\<br />
{x < 0} & \Rightarrow & {x\left| x \right| = - x^2 } \\<br />
\end{array} ?
    You should take a good look at the graph of {x\left| x \right|}.
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