Answer
$$H'(\theta)=\sin\theta+\theta\cos\theta$$
$$H''(\theta)=2\cos\theta-\theta\sin\theta$$
Work Step by Step
$$H(\theta)=\theta\sin\theta$$
1) Find $H'(\theta)$ $$H'(\theta)=(\theta\sin\theta)'$$
Apply the Product Rule: $$H'(\theta)=(\theta)'\sin\theta+\theta(\sin\theta)'$$ $$H'(\theta)=1\times\sin\theta+\theta\cos\theta$$ $$H'(\theta)=\sin\theta+\theta\cos\theta$$
2) Find $H''(\theta)$ $$H''(\theta)=(\sin\theta+\theta\cos\theta)'$$ $$H''(\theta)=(\sin\theta)'+(\theta\cos\theta)'$$
Again, apply the Product Rule: $$H''(\theta)=(\cos\theta)+((\theta)'\cos\theta+\theta(\cos\theta)')$$ $$H''(\theta)=\cos\theta+\cos\theta-\theta\sin\theta$$ $$H''(\theta)=2\cos\theta-\theta\sin\theta$$
