Question

Find the derivatives of the following:

Find the derivative of `sin^-1 ((2x)/(1 + x^2))` with respect to `tan^-1 x`

Answer 1

Let u = `sin^-1 ((2x)/(1 + x^2))`

Put x = tan θ

u = `sin^-1 ((2tantheta)/(1 + tan^2theta))`

u = `sin^-1 (sin 2theta)`

u = 2θ

u = `2 tan^-1 (x)`

`("du")/("d"x) = 2/(1 + x^2)`     ...(1)

Let v = `tan^-1 x`

`("dv")/("d"x) = 2/(1 + x^2)`     ...(2)

From equations (1) and (2)

`(("du")/(dx))/(("dv")/(dx)) = (2/(1  +  x^2))/(1/(1  +  x^2))`

`("du")/("dv")` = 2

`("d"(sin^-1 ((2x)/(1  +  x^2))))/("d"(tan^-1 x))` = 2