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: 61

Answer

$y'=(x^{2}+2)^{2}(x^{4}+4)^{4}[\frac{4x}{(x^{2}+2)}+\frac{16x^{3}}{(x^{4}+4)}]$

Work Step by Step

Use logarithmic differentiation to find the derivative of the function. $y=(x^{2}+2)^{2}(x^{4}+4)^{4}$ Taking logarithmic 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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