Question

Find `dy/dx`, if y = `sin^-1 x + sin^-1 sqrt (1 - x^2)`, 0 < x < 1.

Answer 1

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

Let u = sin⁻¹ x and v = `sin^-1 sqrt (1-x^2)`

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

Now v = `sin^-1 sqrt(1 - x^2)`

Put x = cos θ

∴ v = `sin^-1 sqrt (1 - cos^2 θ)`

= `sin^-1 sqrt (sin^2 θ)`

= sin⁻¹ (sin θ)

= θ

= cos⁻¹ x

∴ `(dv)/dx = -1/sqrt(1 - x^2)`

As `dy/dx = (du)/dx + (dv)/dx`

`= 1/ sqrt(1-x^2) + -1/ sqrt (1 - x^2)`

= 0