Answer
If $v$ has a constant value $c$, then using the Derivative Product Rule, the Derivative Constant Multiple Rule is proved.
Work Step by Step
The Derivative Product Rule states for $u$ and $v$ differentiable at $x$, then $$\frac{d}{dx}(uv)=u\frac{dv}{dx}+v\frac{du}{dx}$$
Now if $v$ has a constant value $c$, or taken directly, $v=c$, then $$\frac{d}{dx}(uc)=u\frac{dc}{dx}+c\frac{du}{dx}$$
Since $c$ is a constant value, $dc/dx=0$. $$\frac{d}{dx}(uc)=u\times0+c\frac{du}{dx}=c\frac{du}{dx}=cu'$$
This, in fact, has proved the Derivative Constant Multiple Rule.
