Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 3 - Section 3.6 - Derivatives of Logarithmic Functions - 3.6 Exercises - Page 223: 30


$f'(x) = \frac{1}{x~\cdot~ln~x~\cdot ~ln~ln~x}$ The domain is $~~(e, \infty)$

Work Step by Step

$f(x) = ln~ln~ln~x$ We can differentiate $f(x)$: $f'(x) = \frac{1}{ln~ln~x}\cdot \frac{d}{dx}(ln~ln~x)$ $f'(x) = \frac{1}{ln~ln~x}\cdot \frac{1}{ln~x} \cdot\frac{d}{dx}(ln~x)$ $f'(x) = \frac{1}{ln~ln~x}\cdot \frac{1}{ln~x} \cdot\frac{1}{x}$ $f'(x) = \frac{1}{x~\cdot~ln~x~\cdot ~ln~ln~x}$ When we consider the function $~~ln~ln~ln~x~~$ it is required that $~~ln~ln~x \gt 0$ Then $~~ln~x \gt 1$ Then $~~x \gt e$ The domain is $~~(e, \infty)$
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