Theorem:For any constant $c$ and any differentiable function $f$, $\frac{d}{dx}(cf(x))=c\ \frac{d}{dx}(f(x))$
Proof: This is just a matter of applying the definition of $\frac{d}{dx}$ and then using the Constant Multiple Law for limits.

Let $F(x)=cf(x)$. Then

$\frac{d}{dx}(F(x))=\lim_{h\to 0}\ \frac{1}{h}(F(x+h)\ -\ F(x))$

$=\lim_{h\to 0}\ \frac{1}{h}(cf(x+h)\ -\ cf(x))$

$=\lim_{h\to 0}\ \frac{c}{h}(f(x+h)\ -\ f(x))$

$=c\lim_{h\to 0}\ \frac{1}{h}(f(x+h)\ -\ f(x))$

$=c\ \frac{d}{dx}(f(x))$