1. The problem asks to find the derivative of $\cos(ax)$ with respect to $x$. 2. The formula for the derivative of $\cos(ax)$ is given as: $$\frac{d}{dx}(\cos(ax)) = -a \sin(ax)$$ 3. Here, $a$ is a constant and $x$ is the variable. 4. This formula comes from the chain rule, which states that the derivative of a composite function $f(g(x))$ is $f'(g(x)) \cdot g'(x)$. 5. In this case, $f(u) = \cos u$ and $g(x) = ax$, so $f'(u) = -\sin u$ and $g'(x) = a$. 6. Applying the chain rule: $$\frac{d}{dx}(\cos(ax)) = -\sin(ax) \cdot a = -a \sin(ax)$$ 7. Therefore, the derivative of $\cos(ax)$ with respect to $x$ is: $$\boxed{-a \sin(ax)}$$