Differentiability of product/composition of function

How will be the product and composition of two functions, where one is differentiable and another is just continuous, behave?I mean to say, if the product or composition is differentiable, then what are the conditions on the functions to hold for their product to be differentiable, if they are?

Some trivial checks: Of course, the product/composition is not always differentiable since if we take the differentiable function to be I (or x), then the result is obviously not differentiable. So what I ask for is that when they do; why?

Let $f,g\colon \Bbb R \to \Bbb R$ and define $h_p,h_c\colon \Bbb R \to \Bbb R$ as the product and the composition of $g$ and $f$, i.e. $$h_p(x)=g(x)f(x),\qquad h_c(x)=g\circ f(x), \qquad \forall x \in \Bbb R.$$

Certainly:

• If $f$ and $g$ are differentiable at $x_0\in \Bbb R$, then $h_p$ is differentiable at $c$.
• If $f$ is differentiable at $x_0\in \Bbb R$ and $g$ is differentiable at $f(x_0)$, then $h_c$ is differentiable at $x_0$.

As shown by the following examples, the reverse of these statements is not true.

Examples:

• $f,g$ not differentiable at $0$, and $h_p$ differentiable.
  Ex: $f(x)=g(x)=|x|$, and $h_p(x)=x^2$

• $f,g$ not differentiable at $0$, and $h_c$ differentiable.
  Ex: $f(x)=\begin{cases} x &\text{if }x\geq 0\\ 2x &\text{if }x<0\end{cases}, g(x)=f(-x)$, and $h_c(x)=2x$:

• $f$ not differentiable at $0$ and $g,h_p,h_c$ differentiable.
  Ex: $f(x)=|x|, g(x)=0$, and $h_c(x)=h_p(x)=0$: