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: 57

Answer

\[\frac{{dy}}{{dx}} = x{\cos ^2}x - x{\sin ^2}x + \cos x\sin x\]

Work Step by Step

\[\begin{gathered} y = x\cos x\sin x \hfill \\ \hfill \\ by\,\,the\,\,product\,\,rule \hfill \\ \hfill \\ \frac{{dy}}{{dx}} = x\,\left( {\cos x\cos x + \sin x\,\left( { - \sin x} \right)} \right) + \cos x\sin x \hfill \\ \hfill \\ {\text{Therefore}}{\text{,}} \hfill \\ \hfill \\ \frac{{dy}}{{dx}} = x\,\left( {{{\cos }^2}x - {{\sin }^2}x} \right) + \cos x\sin x \hfill \\ \hfill \\ simplify \hfill \\ \hfill \\ \frac{{dy}}{{dx}} = x{\cos ^2}x - x{\sin ^2}x + \cos x\sin x \hfill \\ \end{gathered} \]
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