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: 19

Answer

\[\frac{{dy}}{{dx}} = {e^{ - x}}\cos x - {e^{ - x}}\sin x\]

Work Step by Step

\[\begin{gathered} y = {e^{ - x}}\,\sin x \hfill \\ \hfill \\ use\,\,the\,\,product\,\,rule \hfill \\ \hfill \\ \frac{{dy}}{{dx}} = {e^{ - x}}\,{\left( {\sin x} \right)^,} + \,\sin x{\left( {{e^{ - x}}} \right)^,} \hfill \\ \hfill \\ then \hfill \\ \hfill \\ \frac{{dy}}{{dx}} = {e^{ - x}}\,\left( {\cos x} \right) + \sin x\,\left( { - {e^{ - x}}} \right) \hfill \\ \hfill \\ simplify \hfill \\ \hfill \\ \frac{{dy}}{{dx}} = {e^{ - x}}\cos x - {e^{ - x}}\sin x \hfill \\ \hfill \\ \end{gathered} \]
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