Single Variable Calculus, Chapter 3, 3.4, Section 3.4, Problem 14

Differentiate $\displaystyle y = \csc \theta (\theta + \cot \theta)$


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\begin{equation}
\begin{aligned}

y' =& (\csc \theta) \frac{d}{d\theta} (\theta + \cot \theta) + (\theta + \cot \theta) \frac{d}{d\theta} (\csc \theta)
&& \text{Using Product Rule}
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y' =& (\csc \theta)(1 - \csc^2 \theta) + (\theta + \cot \theta)(- \csc \theta \cot \theta)
&& \text{Using Trigonometric Identities}
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\\
y' =& (\csc \theta) (- \cot ^ 2 \theta) + (\theta + \cot \theta) (- \csc \theta \cot \theta)
&& \text{Simplify the equation}
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\\
y' =& - \csc \theta \cot^2 \theta - \theta \csc \theta \cot \theta - \csc \theta \cot^2 \theta
&& \text{Combine like terms}
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y' =& -2 \csc \theta \cot^2 \theta - \theta \csc \theta \cot \theta
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& \text{or}
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y' =& - \csc \theta \cot \theta (2 \cot \theta + \theta)
&& \text{}

\end{aligned}
\end{equation}
$