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Differentaite w.r.t x , y = tan^(-1) (x...

Differentaite w.r.t x , ` y = tan^(-1) (x /(sqrt(1+x^(2))-1))`

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`y = tan^(-1)(x/(sqrt(1+x^(2)-1))) , " put " theta = tan^(-1) x`
` - pi/2 lt theta lt pi/2`
` y = tan^(-1) ((tan theta)/ (sec theta-1))`
` = tan^(-1) ((sin tehta)/(1-cos theta))`
`= tan^(-1) ((2 "sin" theta/2 "cos" theta/2)/(2 sin^(2) (theta)/2))`
`=tan^(-1)(cot ""(theta/2))= {:{(pi/2-theta/2, 0 lt theta lt pi/2), (pi/2-(theta/2 +pi) , -pi/2 lt theta lt 0):}}`
` or -pi/2 - (tan^(-1)x)/2`
` y = pi/2 - (tan^(-1)x)/2`
` (dy)/(dx)= 1/(2(1+x^(2))`
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