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Question
Differentiate the following w.r.t. x : `tan^-1[sqrt((sqrt(1 + x^2) + x)/(sqrt(1 + x^2) - x))]`
Solution
Let y = `tan^-1[sqrt((sqrt(1 + x^2) + x)/(sqrt(1 + x^2) - x))]`
Put x = tanθ. Thenθ = tan–1x
∴ `(sqrt(1 + x^2) + x)/(sqrt(1 + x^2 - x)) = sqrt(1 + tan^2θ + tanθ)/(sqrt(1 + tan^2θ) - tanθ)`
= `"secθ + tanθ"/"secθ - tanθ"`
= `((1/cosθ) + (sinθ/cosθ))/((1/cosθ) - (sinθ/cosθ))`
= `(1 + sinθ)/(1 - sinθ)`
= `(1 - cos(pi/2 + θ))/(1 + cos(pi/2 + θ)`
= `(2sin^2(pi/4 + θ/2))/(2cos^2(pi/4 + θ/2)`
= `tan^2(pi/4 + θ/2)`
∴ `sqrt((sqrt(1 + x^2) + x)/(sqrt(1 + x^2) - x)`
= `tan(pi/4 + θ/2)`
∴ y = `tan^-1[tan(pi/4 + θ/2)]`
= `pi/4 + θ/2`
= `pi/4 + 1/2tan^-1x`
∴ `"d"/"dx"(pi/4) + (1)/(2)"d"/"dx"(tan^-1x)`
= `0 + (1)/(2) xx (1)/(1 + x^2)`
= `(1)/(2(1 + x^2)`.
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