Question

Differentiate the following w.r.t. x : `cos^-1((sqrt(1 + x) - sqrt(1 - x))/2)`

Answer 1

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

Put x = cosθ. Thenθ = cos^–1x and

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

= `((sqrt(1 + cosθ) - sqrt(1 - cosθ))/2)`

= `(sqrt(2cos^2(θ/2)) - sqrt(2sin^2(θ/2)))/2`

= `(cos(θ/2) - sin(θ/2))/sqrt(2)`

= `[cos  θ/(2)](1/sqrt(2)) - [sin  θ/2](1/sqrt(2))`

= `cos  θ/(2).cos  pi/4 - sin  θ/2.sin  pi/4`

= `cos(θ/2 + pi/4)`

∴ y = `cos^-1[cos(θ/2 + pi/4)]`

= `θ/(2) + pi/(4)`

= `(1)/(2)cos^-1x + pi/(4)`

∴ `"dy"/"dx" = (1)/(2)"d"/"dx"(cos^-1x) + "d"/"dx"(pi/4)`

= `(1)/(2) xx (-1)/sqrt(1 - x^2) + 0`

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