# Thread: derivative of |x|

1. ## 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

2. ## hi

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.

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

yes, thats is the question. but i dont get what to do from there... :\

4. 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)

5. 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 $\displaystyle {x\left| x \right|}$.