Question

Differentiate the following w.r.t. x : `sin^-1(2xsqrt(1 - x^2))`

Answer 1

Let y = `sin^-1(2xsqrt(1 - x^2))`
Put x = sinθ.
Then θ = sin^–1x
∴ y = `sin^-1(2sinθsqrt(1 - sin^2θ))`
= sin^–1(2sinθ cosθ)
= sin^–1(sin2θ)
= 2θ
= 2sin^–1x
Differentiating w.r.t. x, we get
`"dy"/"dx" = "d"/"dx"(2sin^-1x)`

= `2"d"/"dx"(sin^-1x)`

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

= `2/sqrt(1 - x^2)`
We can also put x = cosθ.
Then θ = cos^–1x
∴ y = `sin^-1(2cosθsqrt(1 - cos^2θ))`
= sin^–1(2cosθ sinθ)
= sin^–1(sin2θ)
= 2θ
= 2cos^–1x
Differentiating w.r.t. x, we get
`"dy"/"dx" = "d"/"dx"(2cos^-1 x)`

= `2"d"/"dx"(cos^-1 x)`

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

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

Hence, `"dy"/"dx" = ± (2)/sqrt(1 - x^2)`.