Filling in the derivative of the absolute value at zero

Question

I have a function $f(x)$ such that $f(x_0)=0$ and I'm interested in the derivative $\frac{d |f(x)|}{dx}$ evaluated at the point $x_0$.

I realize that this is usually undefined.

However, if $\lim_{x\rightarrow x_0^+}\frac{d |f(x)|}{dx}=0$ and $\lim_{x\rightarrow x_0^-}\frac{d |f(x)|}{dx}=0$, is it ok to say that $\frac{d |f(x)|}{dx}=0$ then?

Answer 1

It is not the case that $\frac{d|f(x)|}{dx}$ is usually undefined at zeros of the function $f$. For instance, if $f(x) = x^2$, then $|f(x)| = f(x)$, hence $|f(x)|$ is clearly differentiable at $x=0$.

So yes, if the limit of the difference quotient defining the derivative of $|f(x)|$ exists, then $|f(x)|$ is differentiable at that point.