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: 12

Answer

\[{y^,} = - 18x{e^{3{x^2} + 5}}\]

Work Step by Step

\[\begin{gathered} y = - 3{e^{3{x^2} + 5}} \hfill \\ Find\,\,the\,\,derivative \hfill \\ {y^,} = {\left( { - 3{e^{3{x^2} + 5}}} \right)^,} \hfill \\ {y^,} = - 3\,{\left( {{e^{3{x^2} + 5}}} \right)^,} \hfill \\ Use\,\,the\,\,formula \hfill \\ \frac{d}{{dx}}\,\,\left[ {{e^{g\,\left( x \right)}}} \right] = {e^{g\,\left( x \right)}}{g^,}\,\left( x \right) \hfill \\ Then \hfill \\ {y^,} = - 3\,\left( {{e^{3{x^2} + 5}}} \right)\,{\left( {3{x^2} + 5} \right)^,} \hfill \\ {y^,} = - 3\,\left( {{e^{3{x^2} + 5}}} \right)\,\,\left( {6x} \right) \hfill \\ Multiplying\, \hfill \\ {y^,} = - 18x{e^{3{x^2} + 5}} \hfill \\ \end{gathered} \]
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