Question

Find `bb(dy/dx)` in the following:

y = `sin^(-1)(2xsqrt(1-x^2)), -1/sqrt2 < x < 1/sqrt2`

Answer 1

y = `sin^-1 (2x sqrt(1 - x^2))`

Let, x = sin θ

⇒ θ = sin⁻¹ x

∴ y = `sin^-1 (2 sin theta sqrt(1 - sin^2 theta))`

= sin⁻¹ (2 sin θ × cos θ)

= sin⁻¹ (sin 2 θ)  ....(∵ sin 2 θ = 2 sin × cos θ)

= 2 θ 

= 2 sin⁻¹ x

On differentiating with respect to x,

`dy/dx = 2 d/dx sin^-1 x`

`dy/dx = 2 xx 1/(sqrt(1 - x^2))`

`dy/dx= 2/sqrt(1 - x^2)`