Question

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

Answer 1

Let u = `tan^-1((sqrt(1 + x^2) - 1)/(x))`
and
v = `tan^-1((2xsqrt(1 - x^2))/(1 - 2x^2))`.

Then we want to find `"du"/"dv"`

u = `tan^-1((sqrt(1 + x^2) - 1)/(x))`
Put x = tanθ.
Thenθ = tan^–1 x
and
`(sqrt(1 ++ x^2) - 1)/(x) = (sqrt(1 + tan^2θ) - 1)/tanθ`

= `(secθ - 1)/(tanθ)`

= `((1)/(cosθ) - 1)/((sinθ/cosθ)`

= `(1 - cosθ)/(sinθ)`

= `(2sin^2(θ/2))/(2sin(θ/2)cos(θ/2))`

= `tan(θ/2)`

∴ u = `tan^-1[tan(θ/2)] = θ/(2) = (1)/(2)tan^-1x`

∴ `"du"/"dx" = (1)/(2)"d"/"dx"(tan^-1x)`

= `(1)/(2) xx (1)/(1 + x^2)`

= `(1)/(2(1 + x^2)`

v = `tan^-1((2xsqrt(1 - x^2))/(1 - 2x^2))`
Put x = sinθ.
Thenθ = sin^–1x
and
`(2xsqrt(1 - x^2))/(1 - 2x^2)`

= `(2sinθsqrt(1 - sin^2θ))/(1 - 2sin^2θ)`

= `(2sinθcosθ)/(1 - 2sin^2θ)`

= `(sin2θ)/(cos2θ)`
= tan2θ
∴ v = tan^-1(tan2θ)
= 2θ
= 2sin^-1x
∴ `"dv"/"dx" = 2"d"/"dx"(sin^-1x)`

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

∴ `"dv"/"dx" = (("du"/"dx"))/(("dv"/"dx")`

= `([(1)/(2(1 + x^2))])/(((2)/sqrt(1 - x^2))`

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

= `sqrt(1 - x^2)/(4(1 + x^2)`.