2015-03-08
A composition of differentiable functions can be non-differentiable.
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 \(x=0\).