1. ## Chain Rule Question

This one comes in two parts, but is basically the same question:

a) If $f(x)= \rvert \sin x \lvert$, find $f'(x)$ and sketch the graphs of $f$ and $f'$. Where is $f$ not differentiable?

b) a) If $g(x)= \sin \rvert x \lvert$, find $g'(x)$ and sketch the graphs of $g$ and $g'$. Where is $f$ not differentiable?

So, I believe that $f'(x)= \rvert \cos x \lvert$ and $g'(x)= \cos \rvert x\lvert$, and know how to graph each of these functions. What I am not confident on is the part about where are these functions not differentiable? As I look at the graphs, it looks to me like they are both differentiable across all real numbers. Am I missing something?

2. They are not differentiable across all real numbers becasue they have sharp corner curves or 'cusps'.

For the first function these appear at the zeros of $\sin x$

Do you follow?

3. I follow the rule of not being differentiable at sharp corners. Maybe my graph is too small to tell, but I didn't realize that there were sharp corners at the zeros of $\vert \sin x\vert '$

Thanks for your help. I appreciate it.