Question

Find the derivative of \( \sec^{-1}(x) \): \[ \frac{d}{dx} \left( \sec^{-1}(x) \right) \]

• A. \( \frac{1}{\sqrt{1 - x^2}} \)
• B. \( \frac{x}{\sqrt{x^2 - 1}} \)
• C. \( \frac{1}{\sqrt{1 + x^2}} \)
• D. \( \frac{-x}{\sqrt{x^2 - 1}} \)

Hint:

The derivative of \( \sec^{-1}(x) \) is \( \frac{1}{|x|\sqrt{x^2 - 1}} \). Remember to use the absolute value of \( x \) to account for the domain restrictions of the inverse secant function.

Answer 1

The Correct Option is B

To differentiate \( \sec^{-1}(x) \), we use the standard formula for the derivative of the inverse secant function: \[ \frac{d}{dx} \left( \sec^{-1}(x) \right) = \frac{1}{|x|\sqrt{x^2 - 1}} \] Thus, the derivative is: \[ \frac{x}{\sqrt{x^2 - 1}} \] Therefore, the correct option is: \[ \boxed{\frac{x}{\sqrt{x^2 - 1}}} \]