Question

Differentiate `sec^-1 (1/sqrt(1 - x^2))` w.r.t. `sin^-1 (2xsqrt(1 - x^2))`.

Answer 1

Let u = `sec^-1  1/sqrt(1 - x^2)`

x = sin θ

`\implies 1/sqrt(1 - x^2) = 1/sqrt(1 - sin^2θ)`

= `1/cosθ`

= sec θ

So u = sec^–1 sec θ

= θ

= sin^–1 x

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

v = `sin^-1 (2xsqrt(1 - x^2))`  ...(i)

Suppose that

x = sin α

v = `sin^-1 [2 sin αsqrt(1 - sin^2α)]`

= sin^–1 [2 sin α . cos α]

= sin^–1 [sin 2α]

= 2α

= 2 sin^–1 x

`(dv)/dx = 2/sqrt(1 - x^2)`  ...(ii)

From (i) and (ii)

∴ `(du)/(dv) = 1/2`