Calculus with Applications (10th Edition)

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

Chapter 5 - Graphs and the Derivative - Chapter Review - Review Exercises - Page 298: 35

Answer

$${\text{ }}f''\left( x \right) = \frac{{180}}{{{{\left( {3x - 6} \right)}^3}}},\,\,\,\,\,f''\left( 1 \right) = - \frac{{20}}{3},\,\,\,\,f''\left( { - 3} \right) = - \frac{4}{{75}}$$

Work Step by Step

$$\eqalign{ & f\left( x \right) = \frac{{4x + 2}}{{3x - 6}} \cr & {\text{find the derivative of }}f\left( x \right) \cr & f'\left( x \right) = \frac{d}{{dx}}\left[ {\frac{{4x + 2}}{{3x - 6}}} \right] \cr & {\text{by using the quotient rule }}\frac{d}{{dx}}\left[ {\frac{{u\left( x \right)}}{{v\left( x \right)}}} \right] = \frac{{v\left( x \right) \cdot u'\left( x \right) - u\left( x \right)v'\left( x \right)}}{{{{\left[ {v\left( x \right)} \right]}^2}}} \cr & f'\left( x \right) = \frac{{\left( {3x - 6} \right)\frac{d}{{dx}}\left[ {4x + 2} \right] - \left( {4x + 2} \right)\frac{d}{{dx}}\left[ {3x - 6} \right]}}{{{{\left( {3x - 6} \right)}^2}}} \cr & {\text{solve derivatives}} \cr & f'\left( x \right) = \frac{{\left( {3x - 6} \right)\left( 4 \right) - \left( {4x + 2} \right)\left( 3 \right)}}{{{{\left( {3x - 6} \right)}^2}}} \cr & {\text{simplifying}} \cr & f'\left( x \right) = \frac{{12x - 24 - 12x - 6}}{{{{\left( {3x - 6} \right)}^2}}} \cr & f'\left( x \right) = - \frac{{30}}{{{{\left( {3x - 6} \right)}^2}}} \cr & \cr & {\text{by using the quotient rule}} \cr & f''\left( x \right) = - \frac{{{{\left( {3x - 6} \right)}^2}\frac{d}{{dx}}\left( {30} \right) - 30\frac{d}{{dx}}\left[ {{{\left( {3x - 6} \right)}^2}} \right]}}{{{{\left( {{{\left( {3x - 6} \right)}^2}} \right)}^2}}} \cr & f''\left( x \right) = - \frac{{ - 30\left( 2 \right)\left( {3x - 6} \right)\frac{d}{{dx}}\left[ {3x - 6} \right]}}{{{{\left( {{{\left( {3x - 6} \right)}^2}} \right)}^2}}} \cr & f''\left( x \right) = - \frac{{ - 30\left( 2 \right)\left( {3x - 6} \right)\left( 3 \right)}}{{{{\left( {{{\left( {3x - 6} \right)}^2}} \right)}^2}}} \cr & {\text{simplifying}} \cr & f''\left( x \right) = \frac{{180}}{{{{\left( {3x - 6} \right)}^3}}} \cr & \cr & {\text{find }}f''\left( 1 \right){\text{ and }}f''\left( { - 3} \right) \cr & f''\left( 1 \right) = \frac{{180}}{{{{\left( {3\left( 1 \right) - 6} \right)}^3}}} = - \frac{{20}}{3} \cr & and \cr & f''\left( { - 3} \right) = \frac{{180}}{{{{\left( {3\left( { - 3} \right) - 6} \right)}^3}}} = - \frac{4}{{75}} \cr} $$
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