Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 3 - Differentiation - 3.1 Definition of the Derivative - Exercises - Page 104: 55


$f(x)=\sin x$ and $a=\frac{\pi}{6}$

Work Step by Step

The definition of derivative at point $a$ is $f'(a)=\lim\limits_{h \to 0}\frac{f(a+h)-f(a)}{h}$ But given that $f'(a)=\lim\limits_{h \to 0}\frac{\sin(\frac{\pi}{6}+h)-0.5}{h}$ $\implies f(a+h)=\sin(\frac{\pi}{6}+h)$ and $f(a)=0.5=\sin \frac{\pi}{6}$ which gives $f(x)=\sin x$ and $a=\frac{\pi}{6}$.
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