Differentiate the following functions with respect to `x`: (i) `tan^(-1){(sqrt(1+x^2)+1)/x},\ \ x!=0` (ii) `cot^(-1){sqrt(1+x^2)+x}`

Explanation

`(i)`Given that,
`y=tan^(-1){(sqrt(1+x^2)+1)/x},\ \ x!=0`
`x=tan theta`
`y=tan^-1[(sec theta +1)/tan theta]`
`=>y=tan^-1 [(1+cos theta)/sin theta ]`
`=>y=tan^-1[(2cos^2(theta/2))/(2 sin(theta/2)*cos(theta/2))]`
`=>y=tan^-1(cot(theta/2))`
`=>y=tan^-1[tan(pi/2-theta/2)]`
`=>y=pi/2-theta/2`
`=>y=pi/2-1/2 tan^-1 x`
`(dy)/(dx)=-1/(2(1+x^2))`



`(ii)`given that,
`y=cot^(-1){sqrt(1+x^2)+x}`
`x=cot theta`
`y=cot^(-1){sqrt(1+cot^2theta)+cot theta}`
`=>y=cot^-1(cosec theta+cot theta)`
`=>y=cot^-1[(1+cos theta)/sin theta]`
`y=cot^-1[(2cos^2(theta/2))/(2 sin(theta/2)*cos (theta/2))]`
`=>y=cot^-1(cot(theta/2))`
`=>y=theta/2`
`y=1/2 cot ^-1 x`
`(dy)/(dx)=1/2*(-1/(1+x^2))`
`=-1/(2(1+x^2))`