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: 38


$$ h'=t^tt^{t^t}\left(\frac{1}{t}+(1+\ln t )\ln t\right).$$

Work Step by Step

To find the derivative, we have $$\ln h= \ln t^{t^t}=t^t\ln t\Longrightarrow \frac{h'}{h}=t^t \frac{1}{t}+(t^t)'\ln t.$$ Now, to find the derivative of $ g=t^t $, we get $$\ln g=t\ln t\Longrightarrow \frac{g'}{g}=\ln t + 1\Longrightarrow g'=t^t(1+\ln t ).$$ Finally, we have $$ h'=t^{t^t}\left(t^t \frac{1}{t}+t^t(1+\ln t )\ln t\right)=t^tt^{t^t}\left(\frac{1}{t}+(1+\ln t )\ln t\right).$$
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