Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 2 - Derivatives - 2.5 The Chain Rule - 2.5 Exercises: 47

Answer

$$y'=-3\sin(\sin3\theta)\cos3\theta$$ $$y''=-9\cos(\sin3\theta)\cos^23\theta+9\sin(\sin3\theta)\sin3\theta$$

Work Step by Step

$$y'=(\cos(\sin3\theta))'=-\sin(\sin3\theta)\cdot(\sin3\theta)'= -\sin(\sin3\theta)\cos3\theta\cdot(3\theta)'= -\sin(\sin3\theta)\cos3\theta\cdot3= -3\sin(\sin3\theta)\cos3\theta$$ $$y''=(y')'=(-3\sin(\sin3\theta)\cos3\theta)'= (-3\sin(\sin3\theta))'\cos3\theta-3\sin(\sin3\theta)(\cos3\theta)'= -3\cos(\sin3\theta)\cdot(\sin3\theta)'\cos3\theta-3\sin(\sin3\theta)(-\sin3\theta)\cdot(3\theta)'= -3\cos(\sin3\theta)\cos3\theta(3\theta)'\cos3\theta+3\sin(\sin3\theta)\sin3\theta\cdot3= -3\cos(\sin3\theta)\cos^23\theta\cdot3+9\sin(\sin3\theta)\sin3\theta=-9\cos(\sin3\theta)\cos^23\theta+9\sin(\sin3\theta)\sin3\theta$$
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