**Proof.**Let F(x) = \sin(2x), f(u) = \sin(u) and g(x) = 2x such that F(x) = f(g(x)). We are going to use the chain rule here:

\begin{align*} F'(x) = f'(g(x))g'(x). \end{align*}

\begin{align*} f'(g(x)) = \cos(2x) \quad \text{and} \quad g'(x) = 2. \end{align*}

\begin{align*} F'(x) &= f'(g(x))g'(x) \\ &= \cos(2x)\cdot 2\\ &= 2\cos(2x). \end{align*}