Calculus with Applications (10th Edition)

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

Chapter 4 - Calculating the Derivative - Chapter Review - Review Exercises: 17

Answer

\[{k^,}\,\left( x \right) = \frac{{21}}{{{{\left( {4x + 7} \right)}^2}}}\]

Work Step by Step

\[\begin{gathered} k\,\left( x \right) = \frac{{3x}}{{4x + 7}} \hfill \\ Find\,\,the\,\,derivative\,\,of\,\,the\,\,function \hfill \\ {k^,}\,\left( x \right) = \frac{d}{{dx}}\,\,\left[ {\frac{{3x}}{{4x + 7}}} \right] \hfill \\ Use\,\,the\,\,quotient\,\,rule \hfill \\ {k^,}\,\left( x \right) = \frac{{\,\left( {4x + 7} \right)\,{{\left( {3x} \right)}^,} - 3x\,{{\left( {4x + 7} \right)}^,}}}{{\,{{\left( {4x + 7} \right)}^2}}} \hfill \\ Then\, \hfill \\ {k^,}\,\left( x \right) = \frac{{\left( {4x + 7} \right)\,\left( 3 \right) - 3x\,\left( 4 \right)}}{{{{\left( {4x + 7} \right)}^2}}} \hfill \\ Simplify\,\,by\,\,multiplying\,\,and\,\,combining\,\,terms\, \hfill \\ {k^,}\,\left( x \right) = \frac{{12x + 21 - 12x}}{{{{\left( {4x + 7} \right)}^2}}} \hfill \\ {k^,}\,\left( x \right) = \frac{{21}}{{{{\left( {4x + 7} \right)}^2}}} \hfill \\ \hfill \\ \end{gathered} \]
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