If $f$ has the constant value $f(x)=c$, then $ \frac{d f}{d x}=\frac{d}{d x}(c)=0 \text {. } $ #### Proof We apply the definition of the derivative to $f(x)=c$, the function whose outputs have the constant value $c$ (Figure 3.9). At every value of $x$, we find that $ f^{\prime}(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 $