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: 75


$$ f'(x) =x^{e^x}(e^x\ln x+(e^x/x))$$

Work Step by Step

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