Derived function

Question

Could the derived function $ f'$ have a domain different from the domain of $f$ ? For example, could we say that the derived function of modulus function is the function $ f' $ : $ \Bbb R $ $\setminus$ {0} $\rightarrow$ $ \Bbb R $ (since the modulus function is not differentiable at $x = 0 $) that is $1$ when $x>0$ and $-1$ when $x<0$ ? If so, is the modulus function of class $C^1$ ? Does it implies that the only functions that are not of class $C^1$ are functions nowhere differentiable? If not, could someone give me a counterexample?