Single Variable Calculus, Chapter 3, 3.4, Section 3.4, Problem 29

Suppose that $H (\theta) = \theta \sin \theta$, find $H'(\theta)$ and $H''(\theta)$

Solving for $H'(\theta)$


$
\begin{equation}
\begin{aligned}

H'(\theta) =& \theta \frac{d}{d \theta} (\sin \theta) + \sin \theta \frac{d}{d \theta} (\theta)
&& \text{Using Product Rule}
\\
\\
H'(\theta) =& (\theta) (\cos \theta) + (\sin \theta) (1)
&& \text{Simplify the equation}
\\
\\
H'(\theta) =& (\theta) (\cos \theta) + (\sin \theta)
&& \text{}
\\
\\

\end{aligned}
\end{equation}
$



Solving for $H''(\theta)$


$
\begin{equation}
\begin{aligned}

H''(\theta) =& \theta \frac{d}{d \theta} (\cos \theta) + \cos \theta \frac{d}{d \theta} (\theta)+ \frac{d}{d \theta} (\sin \theta)
&& \text{Using Product Rule}
\\
\\
H''(\theta) =& \theta (- \sin \theta) + \cos \theta (1) + \cos \theta
&& \text{Simplify the equation}
\\
\\
H''(\theta) =& - \theta \sin \theta + 2 \cos \theta
&& \text{}
\\
\\

\end{aligned}
\end{equation}
$