Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 6 - Inverse Functions - 6.2 Exponential Functions and Their Derivatives - 6.2 Exercises: 59

Answer

$f^{n}(x)=2^{n}e^{2x}$

Work Step by Step

$f(x)=e^{2x}$ and $f'(x)=f^{1}(x)=2e^{2x}$ Again differentiate $f'(x)=f^{1}(x)=2e^{2x}$ with rest to x, we get $f''(x)=f^{2}(x)=2^{2}e^{2x}$ Now, take third derivative. $f'''(x)=f^{3}(x)=2^{3}e^{2x}$ Proceeding this process up to nth derivative. $f^{n}(x)=2^{n}e^{2x}$ Hence, the sequence of nth derivative follows powers of two.
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