# derivative of |x|

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• November 20th 2008, 02:02 PM
bimrocco
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
• November 20th 2008, 02:31 PM
Twig
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.
• November 20th 2008, 02:42 PM
bimrocco
Quote:

Originally Posted by Twig
Uhm, is it meant to say $f(x) = x|x|$

yes, thats is the question. but i dont get what to do from there... :\
• November 20th 2008, 02:46 PM
elizsimca
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) :)
• November 20th 2008, 02:48 PM
Plato
Do you see that $\begin{array}{lcl}
{x > 0} & \Rightarrow & {x\left| x \right| = x^2 } \\
{x < 0} & \Rightarrow & {x\left| x \right| = - x^2 } \\
\end{array}$
?
You should take a good look at the graph of ${x\left| x \right|}$.