Calculus (3rd Edition)

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

Chapter 7 - Exponential Functions - 7.3 Logarithms and Their Derivatives - Exercises - Page 343: 73


$$ f'(x)=x^{3x}(3\ln x+3).$$

Work Step by Step

Recall that $(e^x)'=e^x$ Recall that $(\ln x)'=\dfrac{1}{x}$ We have $$ f(x)=(e^{\ln x})^{3x}=e^{3x\ln x}.$$ Now taking the derivative, we get $$ f'(x)= e^{3x\ln x}(3x\ln x)'=e^{3x\ln x}(3\ln x+3x/x)=e^{3x\ln x}(3x\ln x)'=e^{3x\ln x}(3\ln x+3)=x^{3x}(3\ln x+3).$$
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