Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 7 - Exponential Functions - 7.3 Logarithms and Their Derivatives - Exercises - Page 343: 72


$$ y'=y\left( \frac{3x^2}{ x^3+1}+ \frac{4x^3}{ x^4+2}+\frac{10x^4}{x^5+3}\right).$$

Work Step by Step

Taking the $\ln $ of both sides of the equation, we get $$\ln y= \ln((x^3+1)(x^4+2)(x^5+3)^2)$$ Then using the properties of $\ln $, we have $$\ln y= \ln (x^3+1) +\ln (x^4+2)+2\ln(x^5+3).$$ Now taking the derivative for the above equation, we have $$\frac{y'}{y}= \frac{3x^2}{ x^3+1}+ \frac{4x^3}{ x^4+2}+\frac{10x^4}{x^5+3},$$ Hence, $ y'$ is given by $$ y'=y\left( \frac{3x^2}{ x^3+1}+ \frac{4x^3}{ x^4+2}+\frac{10x^4}{x^5+3}\right).$$
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