The Derivative of a Constant is Zero

Theorem:

If $f\,(x) = c$, for some constant $c$, then $f\,'(x) = 0$.



Proof:

Suppose $f(x)=c$, for some constant $c$. Then the derivative of $f(x)$ can be found as follows

$$\begin{array}{rcl} f\,'(x) & = & \lim_{h \rightarrow 0} \ \frac{f\,(x+h) - f\,(x)}{h}\\\\ & = & \lim_{h \rightarrow 0} \ \frac{c - c}{h}\\\\ & = & \lim_{h \rightarrow 0} \ \ 0\\\\ & = & 0 \end{array}$$

QED.