Calculus with Applications (10th Edition)

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

Chapter 4 - Calculating the Derivative - 4.4 Derivatives of Exponential Functions - 4.4 Exercises: 33

Answer

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

Work Step by Step

\[\begin{gathered} f\,\left( x \right) = {e^{x\sqrt {3x + 2} }} \hfill \\ Find\,\,the\,\,derivative \hfill \\ {f^,}\,\left( x \right) = \,\,\left[ {{e^{x\sqrt {3x + 2} }}} \right] \hfill \\ Use\,\,the\,\,formula \hfill \\ \frac{d}{{dx}}\,\,\left[ {{a^{g\,\left( x \right)}}} \right] = \,\left( {\ln a} \right){a^{g\,\left( x \right)}}{g^,}\,\left( x \right) \hfill \\ Then \hfill \\ {f^,}\,\left( x \right) = {e^{x\sqrt {3x + 2} }}\,\left( {x\sqrt {3x + 2} } \right) \hfill \\ Use\,\,product\,\,rule \hfill \\ {f^,}\,\left( x \right) = {e^{x\sqrt {3x + 2} }}\,\left( {x\,{{\left( {\sqrt {3x + 2} } \right)}^,} + \sqrt {3x + 2} \,{{\left( x \right)}^,}} \right) \hfill \\ Then \hfill \\ {f^,}\,\left( x \right) = {e^{x\sqrt {3x + 2} }}\,\left( {\frac{{3x}}{{2\sqrt {3x + 2} }} + \sqrt {3x + 2} } \right) \hfill \\ \end{gathered} \]
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