Differentiate `tan^(-1){(sqrt(1+x^2)-sqrt(1-x^2))/(sqrt(1+x^2)+sqrt(1-x^2))}` with respect to `cos^(-1)x^2`

Explanation

Let `u= tan ^-1frac{sqrt(1+x^2)-sqrt(1-x^2)}{sqrt(1+x^2)+sqrt(1-x^2)}` and `v=cos^(-1)x^2`
Let `x^2=cos2 theta`
`=>cos2 theta=2cos^2 theta-1`
`=>1+cos2 theta=2cos^2 theta`
`=>1-cos2 theta=2sin^2 theta`
`=>1-2sin^2 theta`
`u=tan^(-1)frac{sqrt(1+cos2 theta)-sqrt(1-cos 2theta)}{sqrt(1+cos2 theta)+sqrt(1-cos 2theta)}`
`v=cos^(-1) cos 2 theta=>2 theta`
`=tan^(-1)frac{cos theta-sin theta}{cos theta+sin theta}`
`=tan^(-1)frac{1-tan theta}{1+tan theta}`
`=tan^(-1)frac{tan frac{pi}{4}-tan theta}{1+tan frac{pi}{4}tan theta}=>tan^(-1)(tan(frac{pi}{4}-theta))`
`u=frac{pi}{4}-theta`
`frac{dv}{d theta}=2 ---(1)`
`frac{du}{d theta}=-1 ---(2)`
`frac{dv}{du}=frac{-1}{2}`