Question

The derivative of `cos^-1 (2x^2 - 1)` w.r.t `cos^-1 x` is ______

Options

• 2

• `(-1)/(2sqrt(1 - x^2))`

• `2/x`

• 1 - x²

Answer 1

The derivative of `cos^-1 (2x^2 - 1)` w.r.t `cos^-1 x` is 2.

Explanation:

Let u = `cos^-1 (2x^2 - 1)` and v = `cos^-1 x`

Putting x = cosθ in both equations, we get

u = `cos^-1(2cos^2theta - 1)`

u = `cos^-1(cos2theta)`

= 2θ

v = `cos^-1(cosθ)`

= θ

∴ `(du)/(d theta) = 2  "and"  (dv)/(d theta) = 1`

∴ `(du)/(dv) = (((du)/(d theta)))/(((dv)/(d theta)))` = 2