Single Variable Calculus, Chapter 3, 3.5, Section 3.5, Problem 43

Determine the derivative of the function $g(x) = (2r \sin rx + n)^p$


$
\begin{equation}
\begin{aligned}
g'(x) & = \frac{d}{dx} (2r \sin rx + n)^p\\
\\
g'(x) & = p(2r \sin rx + n)^{p-1} \frac{d}{dx} (2r \sin rx + n)\\
\\
g'(x) & = p(2r \sin rx + n)^{p-1} \left[ (2r) \frac{d}{dx} ( \sin rx) + \frac{d}{dx} (n) \right]\\
\\
g'(x) & = p(2r \sin rx + n)^{p-1} \left[ (2r) (\cos rx) \frac{d}{dx} (rx) + 0 \right]\\
\\
g'(x) & = p(2r \sin rx + n)^{p-1} \left[ (2r) (\cos rx) (r) \right]\\
\\
g'(x) & = p(2r \sin rx + n)^{p-1} (2r^2 \cos rx)\\
\\
g'(x) & = (2r^2p \cos rx)(2r \sin rx + n)^{p-1}
\end{aligned}
\end{equation}
$