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$ \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

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