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

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

• A. 1
• B. −1
• C. 2
• D. −2

Answer: A) 1

Explanation: u = sin⁻¹(2x√(1−x²)) = 2 sin⁻¹ x (for |x|≤1/√2). v = cos⁻¹(2x²−1) = 2 cos⁻¹ x. du/dx = 2/√(1−x²), dv/dx = −2/√(1−x²). du/dv = −1. At x=1/2, du/dv = −1.