Calculus: Early Transcendentals (2nd Edition)

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

Chapter 3 - Derivatives - 3.5 Derivatives of Trigonometric Functions - 3.5 Exercises: 43

Answer

\[ = 2{e^x}\cos x\]

Work Step by Step

\[\begin{gathered} y = {e^x}\sin x \hfill \\ \hfill \\ differentiate\,\,to\,\,find\,\,y{\,^,} \hfill \\ use\,\,the\,\,product\,\,rule \hfill \\ \hfill \\ y' = \,{\left( {{e^x}} \right)^\prime }\sin x + {e^x}\,{\left( {\sin x} \right)^\prime } \hfill \\ \hfill \\ = {e^x}\sin x + {e^x}\cos x \hfill \\ \hfill \\ differentiate\,\,to\,\,find\,\,{y^,}^, \hfill \\ use\,\,the\,\,product\,\,rule \hfill \\ \hfill \\ y'' = \,{\left( {{e^x}} \right)^\prime }\sin x + {e^x}\,{\left( {\sin x} \right)^\prime } + \,{\left( {{e^x}} \right)^\prime }\cos x + {e^x}\,{\left( {\cos x} \right)^\prime } \hfill \\ \hfill \\ = {e^x}\sin x + {e^x}\cos x + {e^x}\cos x - {e^x}\sin x \hfill \\ \hfill \\ Simplify \hfill \\ \hfill \\ = 2{e^x}\cos x \hfill \\ \hfill \\ \end{gathered} \]
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