What is the derivative of cot inverse(x)?

To find the derivative of the inverse cotangent function, also known as cot inverse or ( \cot^{-1}(x) ), we can utilize the known derivative formulas for inverse trigonometric functions. The derivative of ( \cot^{-1}(x) ) is given specifically by the formula:

[

\frac{d}{dx} \cot^{-1}(x) = -\frac{1}{1+x^2}

]

This formula arises from applying the chain rule and implicit differentiation, considering the relationship between the cotangent function and its inverse.

The reasoning behind the negative sign is that ( \cot^{-1}(x) ) is a decreasing function. As ( x ) increases, ( \cot^{-1}(x) ) decreases, which reflects in the negative derivative.

Now, aligning this with the available choices, the expression ( -\frac{1}{1+x^2} ) confirms that the derivative of ( \cot^{-1}(x) ) matches perfectly with the correct choice provided in the options. Thus, the justification for the correctness of the answer lies in the established derivative rule for the inverse cotangent function.