The derivative of sec⁻¹(1/(2x² − 1)) with respect to √(1 − x²) at x = 1/2 is:

The derivative of sec⁻¹(1/(2x² − 1)) with respect to √(1 − x²) at x = 1/2 is:

• A. 4
• B. −4
• C. 1/4
• D. −1/4

Answer: A) 4

Explanation: Let u = sec⁻¹(1/(2x²−1)) = cos⁻¹(2x²−1) = 2 cos⁻¹ x for x in [0,1]. v = √(1−x²). du/dx = −2/√(1−x²), dv/dx = −x/√(1−x²). So du/dv = 2/x. At x = 1/2, du/dv = 4.