The function $f(x)=|x|$ has a derivative at $x=0$?

Question

The function $f(x)=|x|$ has a derivative at $x=0$?

I already know about $|x|$ is not differentiable at zero. But I can't solve if $|x|$ has a derivative at $x=0$. When $\lim_{h\to 0^+}$, a derivative is $1$, and $\lim_{h \to 0^-}$, a derivative is $-1$.

How about at $x=0$? Help me.

Answer 1

Your question shows that you actually know all the correct things about the absolute value function in order to answer for yourself. It does have left and right derivatives at $0$. In order for the function to be differentiable there these would have to be equal, and they're not.

Answer 2

It is simply not defined at $x=0$, since the one side limit exist but are different the limit at $x=0$ doesn't exist.