1. **State the problem:** Find the derivative of the function $y = \cos x$. 2. **Recall the formula:** The derivative of $\cos x$ with respect to $x$ is given by $$\frac{d}{dx}(\cos x) = -\sin x.$$ This is a fundamental rule in calculus for trigonometric functions. 3. **Apply the formula:** Since our function is exactly $\cos x$, its derivative is $$y' = -\sin x.$$ 4. **Interpretation:** This means that the rate of change of the cosine function at any point $x$ is the negative sine of $x$. For example, at $x=0$, $y' = -\sin 0 = 0$, indicating a horizontal tangent. 5. **Summary:** The derivative of $y = \cos x$ is $$\boxed{y' = -\sin x}.$$