Constant multiple rule of derivatives/Proof

${\frac {d}{dx}}a\,f(x)=a\,f'(x)$, for every constant a.

Prerequisites

Limit definition of the derivative, $f'(x)=\lim_{h \to 0}\frac{f(x+h)-f(x)}{h}$

Proof

Let $g(x)=a\,f(x)$ for some constant a. By the limit definition of the derivative:

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

$a\,f'(x)=a\,\lim _{h\to 0}{\frac {f(x+h)-f(x)}{h}}=\lim _{h\to 0}{\frac {a\,f(x+h)-a\,f(x)}{h}}$

To prove the proposition, it suffices to show that $g'(x)=a\,f'(x)$.

$g'(x)=\lim _{h\to 0}{\frac {g(x+h)-g(x)}{h}}$

$g'(x)=\lim _{h\to 0}{\frac {a\,f(x+h)-a\,f(x)}{h}}=a\,f'(x)$

QED