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: 65

Answer

$$\frac{dy}{dx} = (x-2)^{x+1}\left({\frac{x+1}{x-2}+\ln(x-2)}\right)$$

Work Step by Step

Logarithmic Differentiation is a method used to take the derivative of functions utlizing the logarithmic derivative of a function. Logarithmss were invented by John Napier in order to make mathematics easier, as it does with logarithmic differentiation: $$\ y = (x-2)^{x+1}$$ $$\ln y = (x+1) \ln (x-2)$$ $$\frac{1}{y} (\frac{dy}{dx}) = (x+1)(\frac{1}{x-2}) + \ln(x-2)$$ $$\frac{dy}{dx} = y[\frac{x+1}{x-2} + \ln(x-2)]$$ $$ = \frac{dy}{dx} = (x-2)^{x+1}({\frac{x+1}{x-2}+\ln(x-2)})$$
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