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Differentiate the following functions wi...

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

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