1. Stating the problem: We are given the function $y = \cos(2x)$ and asked to find its derivative $\frac{dy}{dx}$. 2. Recall the chain rule for differentiation: If $y = \cos(u)$ where $u$ is a function of $x$, then $\frac{dy}{dx} = -\sin(u) \cdot \frac{du}{dx}$. 3. Identify the inner function: Here, $u = 2x$. The derivative of $u$ with respect to $x$ is $\frac{du}{dx} = 2$. 4. Apply the chain rule: $$\frac{dy}{dx} = -\sin(2x) \cdot 2 = -2 \sin(2x).$$ 5. Final answer: The derivative of $y = \cos(2x)$ with respect to $x$ is $$\boxed{-2 \sin(2x)}.$$ This corresponds to option (a).