Derivative of Constant

Theorem
Let $f_c \left({x}\right)$ be the constant function on $\R$, where $c \in \R$.

Then $f_c^{\prime} \left({x}\right) = 0$.

Proof
The function $f_c: \R \to \R$ is defined as $\forall x \in \R: f_c \left({x}\right) = c$.

Thus:

Also see
This is the converse of Zero Derivative implies Constant Function.

Thus we see that $f$ is the constant function iff $\forall x: f^{\prime} \left({x}\right) = 0$.