Let the function be $f(x) = \sec x \cot x$. Step 1: Simplify the function using trigonometric identities. We know that $\sec x = \frac{1}{\cos x}$ and $\cot x = \frac{\cos x}{\sin x}$. $$f(x) = \frac{1}{\cos x} \cdot \frac{\cos x}{\sin x}$$ $$f(x) = \frac{1}{\sin x}$$ $$f(x) = \csc x$$ Step 2: Find the derivative of the simplified function. The derivative of $\csc x$ is $-\csc x \cot x$. $$\frac{d}{dx}(\csc x) = -\csc x \cot x$$ Therefore, the derivative of $\sec x \cot x$ is: $$\boxed{-\csc x \cot x}$$