Chapter 2 - Derivatives - 2.5 The Chain Rule - 2.5 Exercises - Page 158: 47

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

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