Calculus (3rd Edition)

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

Chapter 7 - Exponential Functions - Chapter Review Exercises - Page 386: 58


$$x^{\sqrt{x}}\left(x^{\ln x}\right) \left( \frac{\ln \left(x\right)}{2\sqrt{x}}+\frac{1}{\sqrt{x}}+\frac{2\ln \left(x\right)}{x} \right)$$

Work Step by Step

Given $$y= x^{\sqrt{x}}\left(x^{\ln x}\right)$$ Since \begin{align*} \ln y&=\ln x^{\sqrt{x}}\left(x^{\ln x}\right)\\ &=\ln( x^{\sqrt{x}} )+\ln( x^{\ln x} )\\ &= \sqrt{x} \ln( x )+ (\ln x )^2 \end{align*} Differentiate both sides \begin{align*} \frac{1}{y}y'&= \frac{\ln \left(x\right)}{2\sqrt{x}}+\frac{1}{\sqrt{x}}+\frac{2\ln \left(x\right)}{x}\\ y'&=x^{\sqrt{x}}\left(x^{\ln x}\right) \left( \frac{\ln \left(x\right)}{2\sqrt{x}}+\frac{1}{\sqrt{x}}+\frac{2\ln \left(x\right)}{x} \right) \end{align*}
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