Calculus: Early Transcendentals (2nd Edition)

Published by Pearson
ISBN 10: 0321947347
ISBN 13: 978-0-32194-734-5

Chapter 3 - Derivatives - 3.9 Derivatives of Logarithmic and Exponential Functions - 3.9 Exercises: 41

Answer

\[{f^,}\,\left( x \right) = \frac{3}{2}{x^{\frac{1}{2}}}\,\left( {2x - 3} \right) + 2{x^{\frac{3}{2}}}\]

Work Step by Step

\[\begin{gathered} f\,\left( x \right) = \,\left( {2x - 3} \right){x^{\frac{3}{2}}} \hfill \\ \hfill \\ Use\,\,the\,\,product\,\,rule \hfill \\ \hfill \\ {f^,}\,\left( x \right) = \,\left( {2x - 3} \right)\,{\left( {{x^{\frac{3}{2}}}} \right)^,} + {x^{\frac{3}{2}}}\,{\left( {2x - 3} \right)^,} \hfill \\ \hfill \\ differentiate \hfill \\ \hfill \\ {f^,}\,\left( x \right) = \,\left( {2x - 3} \right)\,\left( {\frac{3}{2}{x^{\frac{1}{2}}}} \right) + {x^{\frac{3}{2}}}\,\left( 2 \right) \hfill \\ \hfill \\ multiply \hfill \\ \hfill \\ {f^,}\,\left( x \right) = \frac{3}{2}{x^{\frac{1}{2}}}\,\left( {2x - 3} \right) + 2{x^{\frac{3}{2}}} \hfill \\ \hfill \\ \end{gathered} \]
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