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$ \frac{d}{dx}a=0 $, for some constant a.

Prerequisites

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

Proof

Let f(x) = a for some constant a. From the limit definition of the derivative:

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

QED

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