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: 21

Answer

\[{p^,}\,\left( t \right) = \frac{{ - \frac{1}{{2\sqrt t }} - \frac{1}{2}\sqrt t }}{{\,{{\left( {t - 1} \right)}^2}}}\]

Work Step by Step

\[\begin{gathered} p\,\left( t \right) = \frac{{\sqrt t }}{{t - 1}} \hfill \\ Use\,\,the\,\,quotient\,\,rule\,\,to\,\,find\,\,{p^,}\,\left( t \right) \hfill \\ {p^,}\,\left( t \right) = \frac{{\,\left( {t - 1} \right)\,{{\left( {\sqrt t } \right)}^,} - \,\left( {\sqrt t \,} \right){{\left( {t - 1} \right)}^,}}}{{\,{{\left( {t - 1} \right)}^2}}} \hfill \\ Then \hfill \\ {p^,}\,\left( t \right) = \frac{{\,\left( {t - 1} \right)\,\left( {\frac{1}{{2\sqrt t }}} \right) - \sqrt t \,\left( 1 \right)}}{{\,{{\left( {t - 1} \right)}^2}}} \hfill \\ Simplify\,\,by\,\,multiplying\,\,and\,\,combining\,\,terms \hfill \\ {p^,}\,\left( t \right) = \frac{{\frac{1}{2}{t^{1/2}} - \frac{1}{{2\sqrt t }} - \sqrt t }}{{\,{{\left( {t - 1} \right)}^2}}} \hfill \\ {p^,}\,\left( t \right) = \frac{{ - \frac{1}{{2\sqrt t }} - \frac{1}{2}\sqrt t }}{{\,{{\left( {t - 1} \right)}^2}}} \hfill \\ \end{gathered} \]
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