Question

Find the derivatives of the following:

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

Answer 1

Let y = `cos[2tan^-1 sqrt((1 - x)/(1 + x))]`

Put x = `cos 2theta`

`("d"x)/("d"theta) = - sin 2theta xx 2`

`("d"x)/("d"theta) = - 2 sin 2theta` .......(1)

y = `cos[2tan^- sqrt((1 - cos 2theta)/(1 + cos theta))]`

y = `cos[2tan^-1 sqrt((2sin^2theta)/(2cos^2theta))]`

y = `cos[2tan^-1 sqrt(tan^2theta)]`

y = `cos[2tan^-1 (tan theta)]`

y = `cos[2theta]`

`("d"y)/("d"theta) = - sin 2theta xx 2`

`("d"y)/("d"theta) = - 2 sin 2theta`   .......(2)

From equation (1) and (2) we get

`(("d"y)/("d"theta))/(("d"x)/("d"theta)) = (- 2sin 2theta)/(- 2 sin 2theta)`

`("d"y)/("d"x)` = 1