Differentiate `tan^(-1)((sqrt(1+x^2-1))/x)` with respect to `tan^(-1)x ,` when `x!=0.`

Explanation

`"Let "tan^(-1)((sqrt(1+x^(2)-1))/(x))and v = tan^(-1)x.`
Putting `x=tan theta,` we get
`u=tan^(-1)((sqrt(1+x^(2)-1))/(x))`
`=tan^(-1)((sec theta-1)/(tan theta))`
`=tan^(-1)((1-cos theta)/(sin theta))`
`=tan^(-1)(tan""(theta)/(2))`
`=(1)/(2)theta`
`=(1)/(2)tan^(-1)x`
Thus, we have `u=(1)/(2)tan^(-1)x" and "v=tan^(-1)x`. Therefore,
`(du)/(dx)=(1)/(2)xx(1)/(1+x^(2))and (dv)/(dx)=(1)/(1+x^(2))`
`therefore" "(du)/(dv)=(du//dx)/(dv//dx)=(1)/(2(1+x^(2)))(1+x^(2))=(1)/(2)`