1. ## Chain Rule Question

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

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

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

So, I believe that $\displaystyle f'(x)= \rvert \cos x \lvert$ and $\displaystyle g'(x)= \cos \rvert x\lvert$, 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 $\displaystyle \sin x$

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 $\displaystyle \vert \sin x\vert '$

Thanks for your help. I appreciate it.