What is the derivative of the inverse cosecant function?

The derivative of the inverse cosecant function, written as ( \text{csc}^{-1}(x) ) or ( \arccsc(x) ), is given by the formula:

[

\frac{d}{dx} \left( \arccsc(x) \right) = -\frac{1}{|x|\sqrt{x^2-1}}

]

This formula indicates that the derivative is negative, which reflects that the inverse cosecant function is decreasing in its range. Additionally, the expression contains the term ( \sqrt{x^2 - 1} ) in the denominator, which is derived from the properties of the cosecant function and its relationship to the unit circle, where ( x ) must be greater than or equal to 1 or less than or equal to -1 for the function to be defined.

The presence of the absolute value in the derivative is crucial as it ensures the derivative is defined for both branches of the inverse cosecant (the positive and the negative), reflecting the fact that ( x ) can take either sign but must always lie outside the interval (-1, 1).

Thus, the correct answer appropriately captures the general form of the derivative, aligning it with