The derivative of cos ^–1 (2x^2 – 1) with respect to cos^–1 x is A. 2 B. 1/(2√(1-x^2)) C. 2/x D. 1 - x^2

Question

The derivative of cos^–1(2x² – 1) with respect to cos^–1 x is

A. 2

B. \(\cfrac{1}{2\sqrt{1-\text x^2}}\)

C. \(\cfrac{2}{\text x}\)

D. 1 - x²

Answer 1

  Correct option is A. 2

Let u = cos^–1 (2x² – 1) and v = cos^–1 x

\(\cfrac{du}{dv}=?\)

Considering u = cos^–1 (2x² – 1)

Put x = cos θ ⇒ θ = cos^-1x ---(1)

u = cos^–1 (2cos²θ – 1)

u = cos^–1 (cos2θ)

∵ 2cos²θ – 1 = cos2θ

u = 2θ

u = 2 cos^-1x -- From(1)

Differentiating w.r.t x we get,