Determine where h(x) := x $\lvert x \rvert$ is differentiable from R to R. Determine where h(x) := x $\lvert x \rvert$ is differentiable from R to R.

Not totally sure how to start this. 

Much appreciation,

Jesse
If at all possible. The first thing to do is graph the function and see if it looks differentiable.
 Let $f(x) = x |x|$

Case 1. If $x &gt; 0$, then $f(x) = x^2$, which is differentiable.
So $f(x)$ is differentiable on $(0,\infty)$

Case 2. If $x &lt; 0$, then $f(x) = -x^2$, which is differentiable. So $f(x)$ is differentiable on $(-\infty, 0)$

Case 3. If $x = 0$, then the derivative as $x\rightarrow 0^+=0$ and the derivative as $x\rightarrow 0^-=0$. So $f'(0) = 0$.

Hence f is differentiable on all of $\mathbb R$.
The only possible place where $h(x)$ may fail to be differentiable is $x=0$.  So consider 
$$lim_{x \to 0} \frac{x|x|}{x}.$$
Does this limit exist? If so, $h(x)$ is differentiable everywhere.