A composition of differentiable functions can be non-differentiable.

# Intro

I was a little surprised to find that $$\sqrt{1-\cos(x)}$$, which is a composition of differentiable functions, is not itself differentiable (at e.g. $$x=0$$). A plot illustrates what goes wrong.

Its derivative is $\frac{-\sin(x)}{2\sqrt{1-\cos(x)}}$ which is indeed $$0/0$$ at $$x=0$$.