**Proof.**Let F(x) = \cos(2x), f(u) = \cos(u), and g(x) = 2x such that F(x) = f(g(x)). To prove the derivative of \cos(2x), we will use the chain rule:

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

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

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