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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- Nov 20th 2008, 02:02 PMbimroccoderivative 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 - Nov 20th 2008, 02:31 PMTwighi
Uhm, is it meant to say $\displaystyle f(x) = x|x| $ , and not

$\displaystyle f(x) = |x| $ ?

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

$\displaystyle \lim_{h \rightarrow 0} \frac{f(x+h) - f(x)}{h} $ does not exist in x = 0. - Nov 20th 2008, 02:42 PMbimrocco
- Nov 20th 2008, 02:46 PMelizsimca
I would start this problem by piecewise defining $\displaystyle f(x)=(x)|x|$, 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 $\displaystyle f(x)=(x)|x|$ 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) :) - Nov 20th 2008, 02:48 PMPlato
Do you see that $\displaystyle \begin{array}{lcl}

{x > 0} & \Rightarrow & {x\left| x \right| = x^2 } \\

{x < 0} & \Rightarrow & {x\left| x \right| = - x^2 } \\

\end{array} $?

You should take a good look at the graph of $\displaystyle {x\left| x \right|}$.