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Expert-verified Found in: Page 198 ### Calculus

Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861 # use the definition of derivative to directly prove the differentiation rules for constant and identity function

We use definition of the derivative to prove the differentiation rules for constant and identity function

See the step by step solution

## Step 1: Given information

We are given constant and identity function

## Step 2: Derivative for constant function

Consider a constant function $f:R\to R$ such that $f\left(x\right)=c$ for all $x\in R$

Now we use definition of derivative

$\underset{h\to 0}{\mathrm{lim}}\frac{f\left(x+h\right)-f\left(x\right)}{h}\phantom{\rule{0ex}{0ex}}\underset{h\to 0}{\mathrm{lim}}\frac{c-c}{h}asforanyxwehavef\left(x\right)=c\phantom{\rule{0ex}{0ex}}=0$

## Step 3: Derivative for identity function

Consider the identity function $f:R\to R$ given by $f\left(x\right)=x$

Now we apply the definition of the derivative

$\underset{h\to 0}{\mathrm{lim}}\frac{f\left(x+h\right)-f\left(x\right)}{h}\phantom{\rule{0ex}{0ex}}\underset{h\to 0}{\mathrm{lim}}\frac{x+h-x}{h}\phantom{\rule{0ex}{0ex}}\underset{h\to 0}{\mathrm{lim}}\frac{h}{h}\phantom{\rule{0ex}{0ex}}=1$ ### Want to see more solutions like these? 