Statement
The derivative of a function with a coefficient is the coefficient times the derivative of that function.
$$ \tfrac{d}{dx}\bigg(c*g(x)\bigg) = c * g'(x) $$
Proof
Define function $ f $.
$$ f(x) = c * g(x) $$
Take the definition of the derivative.
$$ f'(x) = \lim_{h \to 0}\left(\frac{f(x + h) - f(x)}{h}\right) $$
$$ f'(x) = \lim_{h \to 0}\left(\frac{c * g(x + h) - c * g(x)}{h}\right) $$
Factor out the $ c $ and move it out the fraction.
$$ f'(x) = \lim_{h \to 0}\left(\frac{c * \bigg(g(x + h) - g(x)\bigg)}{h}\right) $$
$$ f'(x) = \lim_{h \to 0}\left( c * \frac{g(x + h) - g(x)}{h}\right) $$
Move the $ c $ out the limit.
$$ f'(x) = c * \lim_{h \to 0}\left(\frac{g(x + h) - g(x)}{h}\right) $$
Now the definition of the derivative of $ g(x) $ arised.
$$ f'(x) = c * g'(x) $$