Derivative of Constant

Theorem
Let $\map {f_c} x$ be the constant function on $\R$, where $c \in \R$.

Then:
 * $\map { {f_c}'} x = 0$

Proof
The function $f_c: \R \to \R$ is defined as:
 * $\forall x \in \R: \map {f_c} x = c$

Thus:

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

Thus we see that $f$ is the constant function $\forall x: \map {f'} x = 0$.