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