1. The problem asks for the derivative of the inverse cotangent function, $\cot^{-1} x$.\n\n2. Recall the derivative formula for the inverse cotangent function: $$\frac{d}{dx} \cot^{-1} x = -\frac{1}{1+x^2}.$$\n\n3. This means the derivative is negative one divided by the quantity $1 + x^2$.\n\n4. Comparing this with the options given: A is $-\csc^2 x$, B is $\sec^2 x$, C is $-\frac{1}{1+x^2}$, and D is $\frac{1}{1+x^2}$.\n\n5. The correct answer matches option C: $$-\frac{1}{1+x^2}.$$