Calculus 8th Edition

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

Chapter 6 - Inverse Functions - 6.2* The Natural Logarithmic Functions - 6.2* Exercises: 62

Answer

$y'=\frac{(x+1)(x-5)^{3}}{(x-3)^{8}}[\frac{4}{x+1}+\frac{3}{x-5}-\frac{8}{x-3}]$

Work Step by Step

Use logarithmic differentiation to find the derivative of the function. $y=\frac{(x+1)(x-5)^{3}}{(x-3)^{8}}$ Taking logarthimic on both sides . $lny=ln[\frac{(x+1)(x-5)^{3}}{(x-3)^{8}}]$ Use logarithmic properties $ln(xy)=lnx+lny$ , $ln(\frac{x} {y})=lnx-lny$ and $ln(x^{y})=ylnx$. $lny=4ln(x+1)+3ln(x-5)-8ln(x-3)$ Differentiate with respect to $x$. $\frac{1}{y}\frac{dy}{dx}=\frac{4}{x+1}+\frac{3}{x-5}-\frac{8}{x-3}$ $y'=y[\frac{4}{x+1}+\frac{3}{x-5}-\frac{8}{x-3}]$ Hence, $y'=\frac{(x+1)(x-5)^{3}}{(x-3)^{8}}[\frac{4}{x+1}+\frac{3}{x-5}-\frac{8}{x-3}]$
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