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: 39

Answer

\[\frac{{dy}}{{dz}} = {\cos ^2}z - {\sin ^2}z\]

Work Step by Step

\[\begin{gathered} y = \frac{1}{{\sec z\csc z}} \hfill \\ \hfill \\ rewrite\,\,use\,trigonometric\,identities \hfill \\ \hfill \\ y = \cos z\sin z \hfill \\ \hfill \\ {\text{using}}\,\,the\,\,product\,rule. \hfill \\ \hfill \\ \frac{{dy}}{{dz}} = \cos z\,\left( {\cos z} \right) + \sin z\,\left( { - \sin z} \right) \hfill \\ \hfill \\ simplify \hfill \\ \hfill \\ \frac{{dy}}{{dz}} = {\cos ^2}z - {\sin ^2}z \hfill \\ \hfill \\ \end{gathered} \]
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