Is the following derivative application statement true or false?

Determine if the following statement is true or false. Provide proof if true, or a counterexample if false.

Statement:

If $$(f\circ g)(x)$$ is differentiable, then $$f(x)$$ and $$g(x)$$ must be differentiable.

I think that this statement is false and my counterexample is as follows: let $$f(x) = |x|$$, $$g(x) = x^2$$, however, I wanted to find a different example that doesn't make use of $$x$$, $$x^2$$, or any constant functions, but I can't seem to find one.

Could anybody give me a hand, please?

Take any bijection $$f:\mathbb R\to\mathbb R$$, and consider its inverse. Then $$f\circ f^{-1}$$ is the identity function which is obviously differentiable, but $$f$$ need not be.

$$f(x) =g(x)= \begin{cases} 1, & \text{if x is rational} \\ 0, & \text{if x is irrational} \end{cases}$$

I wanted to mention this even though $$f\circ g$$ is constant because it seems cool to me that neither $$f$$ nor $$g$$ are continuous anywhere.

Let $$\text{sign}(x) = 1$$ if $$x >= 0$$, and $$-1$$ if $$x<0$$

Let $$f(x) = -\text{sign}(x)$$

Let $$g(x) = 1$$ if $$x<0$$ else $$g(x)=x$$

Then $$(f \circ g)(x) = 1$$