Calculus 10th Edition

Published by Brooks Cole
ISBN 10: 1-28505-709-0
ISBN 13: 978-1-28505-709-5

Chapter 5 - Logarithmic, Exponential, and Other Transcendental Functions - 5.5 Exercises - Page 363: 67



Work Step by Step

$y=x^{sinx}$ $lny=lnx^{sinx}$ $lny=(sinx)(lnx)$ $\frac{dy}{y}=((cosx)(lnx)+\frac{sinx}{x})dx$ $\frac{dy}{dx}=((cosx)(lnx)+\frac{sinx}{x})x^{sinx}$ at $(\frac{\pi}{2},\frac{\pi}{2}),$ $\frac{dy}{dx}=M=(cos(\frac{\pi}{2})ln(\frac{\pi}{2})+\frac{2sin(\frac{\pi}{2})}{\pi})\frac{\pi}{2}^{sin(\frac{\pi}{2})}$ $=(\frac{2}{\pi})(\frac{\pi}{2})$ $=1$ Equation of Tangent: $y-\frac{\pi}{2}=1(x-\frac{\pi}{2})$ $y=x$
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