Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 6 - Inverse Functions - 6.4 Derivatives of Logarithmic Functions - 6.4 Exercises: 47

Answer

$y'=x^{x}(1+lnx)$

Work Step by Step

Given: $y =x^{x}$ Taking logarithmic on both sides of the function$y =x^{x}$. Use logarithmic property $ln(x^{y})=ylnx$ $lny=xlnx$ Take implicit differentiation with respect to $x$. Apply product rule of differentiation. $\frac{d}{dx}(lny)=\frac{d}{dx}(xlnx)$ $\frac{1}{y}\frac{d}{dx}(y)=x\frac{d}{dx}(lnx)+lnx\frac{d}{dx}(x)$ $\frac{d}{dx}(y)=y[x\times(\frac{1}{x})+lnx\times1]$ Hence, $y'=x^{x}(1+lnx)$
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