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Differentiate tan^(-1){sqrt(1+x^2)+x},\ ...

Differentiate `tan^(-1){sqrt(1+x^2)+x},\ x in RR` with respect to `x` .

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Verified by Experts

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 [(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 cot^-1 x`
`(dy)/(dx)=-1/2(-1/(1+x^2))`
`=1/(2(1+x^2))`
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