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: 15

Answer

\[ = 2{e^{4x}}\,\left( {2{x^2} + 13x + 21} \right)\]

Work Step by Step

\[\begin{gathered} y = \,{\left( {x + 3} \right)^2}{e^{4x}} \hfill \\ Expand\,\,\,\,{\left( {x + 3} \right)^2} \hfill \\ y = \,\left( {{x^2} + 6x + 9\,} \right)\,{e^{4x}} \hfill \\ Find\,\,the\,\,derivative \hfill \\ {y^,} = \,\,\left[ {\,\left( {{x^2} + 6x + 9} \right){e^{4x}}} \right]{\,^,} \hfill \\ Use\,\,the\,\,product\,\,rule\, \hfill \\ {y^,} = \left( {{x^2} + 6x + 9} \right)\,{\left( {{e^{4x}}} \right)^,} + {e^{4x}}{\left( {{x^2} + 6x + 9} \right)^,} \hfill \\ Use\,\,\,\,\frac{d}{{dx}}\,\,\left[ {{e^{g\,\left( x \right)}}} \right] = {e^{g\,\left( x \right)}}{g^,}\,\left( x \right) \hfill \\ Then \hfill \\ {y^,} = \left( {{x^2} + 6x + 9} \right)\,\left( {4{e^{4x}}} \right) + {e^{4x}}\,\left( {2x + 6} \right) \hfill \\ Simplify\,\,by\,multiplying\,\,and\,\,combining\,\,terms \hfill \\ {y^,} = 4{x^2}{e^{4x}} + 24x{e^{4x}} + 36{e^{4x}} + 2x{e^{4x}} + 6{e^{4x}} \hfill \\ {y^,} = 4{x^2}{e^{4x}} + 26x{e^{4x}} + 42{e^{4x}} \hfill \\ Factor \hfill \\ = 2{e^{4x}}\,\left( {2{x^2} + 13x + 21} \right) \hfill \\ \end{gathered} \]
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