Calculus 8th Edition

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

Chapter 6 - Inverse Functions - 6.3* The Natural Exponential Function - 6.3* Exercises: 56

Answer

$y=-(e+1)(x+1)$

Work Step by Step

Find an equation of the tangent line to the curve $xe^{y}+ye^{x}=1$ at the point (0,1). Differentiate $xe^{y}+ye^{x}=1$ with respect to x on both sides. $xe^{y}\frac{dy}{dx}+e^{y}+ye^{x}+e^{x}\frac{dy}{dx}=0$ $y'=-\frac{e^{y}+ye^{x}}{e^{x}+xe^{x}}$ Let M be the slope of the equation at point (0,1). $M=y'|_{(0,1)}=-(e+1)$ Therefore, the equation of the tangent at (0, 1) with slope$M=-(e+1)$ is given as: $y=-(e+1)(x+1)$
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