Calculus with Applications (10th Edition)

Published by Pearson
ISBN 10: 0321749006
ISBN 13: 978-0-32174-900-0

Chapter 4 - Calculating the Derivative - 4.5 Derivatives of Logarithmic Functions - 4.5 Exercises - Page 240: 9

Answer

$$\frac{{dy}}{{dx}} = 3\left( {\frac{{2{x^3} + 5x}}{{{x^4} + 5{x^2}}}} \right)$$

Work Step by Step

$$\eqalign{ & y = \ln {\left( {{x^4} + 5{x^2}} \right)^{3/2}} \cr & {\text{use the power property of logarithms}} \cr & y = \frac{3}{2}\ln \left( {{x^4} + 5{x^2}} \right) \cr & {\text{differentiate with respect to }}x \cr & \frac{{dy}}{{dx}} = \frac{d}{{dx}}\left[ {\frac{3}{2}\ln \left( {{x^4} + 5{x^2}} \right)} \right] \cr & \frac{{dy}}{{dx}} = \frac{3}{2}\frac{d}{{dx}}\left[ {\ln \left( {{x^4} + 5{x^2}} \right)} \right] \cr & {\text{use }}\frac{d}{{dx}}\ln g\left( x \right) = \frac{{g'\left( x \right)}}{{g\left( x \right)}} \cr & \frac{{dy}}{{dx}} = \frac{3}{2}\left( {\frac{{\left( {{x^4} + 5{x^2}} \right)'}}{{{x^4} + 5{x^2}}}} \right) \cr & \frac{{dy}}{{dx}} = \frac{3}{2}\left( {\frac{{4{x^3} + 10x}}{{{x^4} + 5{x^2}}}} \right) \cr & \frac{{dy}}{{dx}} = 3\left( {\frac{{2{x^3} + 5x}}{{{x^4} + 5{x^2}}}} \right) \cr} $$
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