Question

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

Answer 1

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

Then we want to find `"du"/"dv"`.
Put x = tanθ.
Then = tan^–1 x.
∴ u = `cos^-1((1 - tan^2θ)/(1 + tan^2θ))`
= cos^–1(cos2θ)
= 2θ
∴ u = 2tan^–1x
∴ `"du"/"dx" = 2."d"/"dx"(tan^-1x)`

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

= `(2)/(1 + x^2)`
Also, v =tan^–1x
∴ `"dv"/"dx" = "d"/"dx"(tan^-1x) =  (1)/(1 + x^2)`

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

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