Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 3 - Section 3.6 - Derivatives of Logarithmic Functions - 3.6 Exercises - Page 223: 39



Work Step by Step

Use logarithmic differentiation to find the derivative of the function. $y=(x^{2}+2)^{2}(x^{4}+4)^{4}$ Taking the log on both sides. $lny=ln[(x^{2}+2)^{2}(x^{4}+4)^{4}]$ Use logarithmic properties $ln(xy)=lnx+lny$ and $ln(x^{y})=ylnx$. $lny=2ln(x^{2}+2)+4ln(x^{4}+4)$ Differentiate with respect to $x$ $\frac{1}{y}\frac{dy}{dx}=\frac{2}{(x^{2}+2)}\times2x+\frac{4}{(x^{4}+4)}\times4x^{3}$ Hence, $y'=(x^{2}+2)^{2}(x^{4}+4)^{4}[\frac{4x}{(x^{2}+2)}+\frac{16x^{3}}{(x^{4}+4)}]$
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