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