Calculus: Early Transcendentals (2nd Edition)

Published by Pearson
ISBN 10: 0321947347
ISBN 13: 978-0-32194-734-5

Chapter 3 - Derivatives - 3.9 Derivatives of Logarithmic and Exponential Functions - 3.9 Exercises: 63

Answer

\[{y^,} = \frac{{2{x^{\ln x}}\ln x}}{x}\]

Work Step by Step

\[\begin{gathered} f\,\left( x \right) = {x^{\ln x}} \hfill \\ \hfill \\ f\,\left( x \right) = y\,\,\,then\,\,y = {x^{\ln x}} \hfill \\ \hfill \\ then \hfill \\ \hfill \\ \ln y = \ln {x^{\ln x}} \hfill \\ \hfill \\ use\,\,\log \,\,properties \hfill \\ \hfill \\ \ln y = \left( {\ln x} \right)\,\left( {\ln x} \right) \hfill \\ \hfill \\ \ln y = {\ln ^2}x \hfill \\ \hfill \\ Differentiate \hfill \\ \hfill \\ \frac{{{y^,}}}{y} = 2\ln x\,\left( {\frac{1}{x}} \right) \hfill \\ \hfill \\ Solve\,\,for\,\,{y^,} \hfill \\ \hfill \\ {y^,} = 2y\ln x\,\left( {\frac{1}{x}} \right) \hfill \\ \hfill \\ then \hfill \\ \hfill \\ {y^,} = \frac{{2{x^{\ln x}}\ln x}}{x} \hfill \\ \end{gathered} \]
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