Differentiate tan⁻¹(√(1+x²) - x) with respect to x.

d[tan⁻¹(√(1+x²) - x)]/dx = 1/(1 + (√(1+x²) - x)²) × d(√(1+x²) - x)/dx
= 1/(1 + (1+x² + x² - 2x√(1+x²))) × (x/(2√(1+x²)) - 1)
= 1/(1 + 1 + x² - 2x√(1+x²)+x²) × (x - 2√(1+x²))/(2√(1+x²))
= 1/(2 + 2x² - 2x√(1+x²)) × (x - 2√(1+x²))/(2√(1+x²))
= (x - 2√(1+x²))/(2√(1+x²)(2 + 2x² - 2x√(1+x²)))
= (x - 2√(1+x²))/(4√(1+x²)(1 + x² - x√(1+x²)))
= (x - 2√(1+x²))/(4√(1+x²)(1 + x² - x√(1+x²))) × (1 + x² + x√(1+x²))/(1 + x² + x√(1+x²))
= (x + x³ - 2(1+x²) - 2x√(1+x²) + x√(1+x²)(1+x²))/(4√(1+x²)((1+x²)² - x²(1+x²)))
= (x + x³ - 2 - 2x² - x√(1+x²) + x(1+x²)√(1+x²))/(4√(1+x²)(1 + 2x² + x⁴ - x²(1+x²)))
= (x³ -2x² + x - 2 + x(1+x²)√(1+x²) - x√(1+x²))/(4√(1+x²)(1 + x⁴))
= (x³ - 2x² + x - 2 + x(1+x²)√(1+x²) - x√(1+x²))/(4√(1+x²)(1 + x⁴))
= (x³ - 2x² + x - 2 + x(1+x²)√(1+x²) - x√(1+x²))/(4√(1+x²)(1 + x⁴))
After simplification
= 1/(2(1+x²))