What is the derivative of sec inverse(x)?

The derivative of the secant inverse function, sec⁻¹(x), is derived using implicit differentiation or through the application of trigonometric identities and derivatives. The key result for the derivative of sec⁻¹(x) states that:

d/dx[sec⁻¹(x)] = 1 / |x|√(x² - 1).

When considering the result, it is crucial to note that the expression |sqrt(x² - 1)| appears in the denominator. This is because the secant function is defined as the reciprocal of the cosine function, and the inverse secant function is applicable for values where |x| ≥ 1, which leads to the following:

1. When x is greater than or equal to one, the term √(x² - 1) is positive.

2. When x is less than or equal to negative one, |x| ensures that the derivative remains defined in the context of the absolute value.

The correct answer provided aligns with this established mathematical derivative, confirming that it correctly represents the behavior of sec⁻¹(x) in the specified domain. The absolute value of the square root in the denominator ensures that the derivative is always non-negative for valid input values, reflecting