Calculus with Applications (10th Edition)

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

Chapter 4 - Calculating the Derivative - 4.2 Derivatives of Products and Quotients - 4.2 Exercises: 11

Answer

\[{f^,}\,\left( x \right) = \frac{{57}}{{\,{{\left( {3x + 10} \right)}^2}}}\]

Work Step by Step

\[\begin{gathered} f\,\left( x \right) = \frac{{6x + 1}}{{3x + 10}} \hfill \\ Use\,\,the\,\,quotient\,\,rule\,\,to\,\,find\,\,{f^,}\,\left( x \right) \hfill \\ {f^,}\,\left( x \right) = \frac{{\,\left( {3x + 10} \right)\,{{\left( {6x + 1} \right)}^,} - \,\left( {6x + 1} \right)\,{{\left( {3x + 10} \right)}^,}}}{{\,{{\left( {3x + 10} \right)}^2}}} \hfill \\ Then \hfill \\ {f^,}\,\left( x \right) = \frac{{\,\left( {3x + 10} \right)\,\left( 6 \right) - \left( {6x + 1} \right)\,\left( 3 \right)}}{{\,{{\left( {3x + 10} \right)}^2}}} \hfill \\ Simplify\,\,by\,\,multiplying\,\,and\,\,combining\,\,terms \hfill \\ {f^,}\,\left( x \right) = \frac{{18x + 60 - 18x - 3}}{{\,{{\left( {3x + 10} \right)}^2}}} \hfill \\ {f^,}\,\left( x \right) = \frac{{57}}{{\,{{\left( {3x + 10} \right)}^2}}} \hfill \\ \hfill \\ \end{gathered} \]
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