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Q. 84

Expert-verifiedFound in: Page 198

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

We are given constant and identity 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

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

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

Now we apply the definition of the derivative

$\underset{h\to 0}{\mathrm{lim}}\frac{f(x+h)-f\left(x\right)}{h}\phantom{\rule{0ex}{0ex}}\underset{h\to 0}{\mathrm{lim}}\frac{x+h-x}{h}\phantom{\rule{0ex}{0ex}}\underset{h\to 0}{\mathrm{lim}}\frac{h}{h}\phantom{\rule{0ex}{0ex}}=1$

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