Questions about functions with nonzero derivatives almost everywhere

I have two questions about the properties of the functions with nonzero derivatives almost everywhere.

Let $f(x)$ and $g(x)$ be two functions where $f'(x)\neq 0$ and $g'(x)\neq 0$ a.e.

Would $f(x)g(x)$ and $f(g(x))$ have nonzero derivatives a.e. as well?

If there are resources I can use to learn more about it, that would be also awesome!

Edit: Although the question is general, I was curious about the cases for $f(x)$ and $g(x)$ being sigmoid functions.