Single Variable Calculus, Chapter 3, 3.5, Section 3.5, Problem 58

Given the function $f(x) = \sin (x + \sin 2 x); 0 \leq x \leq \pi$ arises in applications to Frequently Modulation (FM) synthesis.

Sketch the graph of $f'$ using a graph of $f$.








b.) Find and graph $f'$.


$\begin{equation} \begin{aligned} f'(x) =& \frac{d}{dx} [\sin (x + \sin 2x)] \\ \\ f'(x) =& \cos (x + \sin 2x) \cdot \frac{d}{dx} (x + \sin 2x) \\ \\ f'(x) =& \cos (x + \sin 2x) \cdot \left[ \frac{d}{dx} (x) + \frac{d}{dx} (\sin 2x) \right] \\ \\ f'(x) =& \cos (x + \sin 2x) [1 + (\cos 2x) (2)] \\ \\ f'(x) =& \cos (x + \sin 2x) (1 + 2 \cos 2 x) \end{aligned} \end{equation}$