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

Differentiate `tan^(-1)((sqrt(1+x^2-1))/x)`withrespect to `tan^(-1)x ,

1.	Differentiate `tan^(-1)((sqrt(1+x^2-1))/x)`withrespect to `tan^(-1)x ,`when `x!=0.`
Answer» Correct Answer - B Let `y = tan^(-1)[(sqrt(1+x^(2))-1)/(x)]and u = tan^(-1)x` Put `x = tan theta rArr theta = tan^(-1)x` Then, `y = tan^(-1)[(sqrt(1+tan^(2)theta)-1)/(tan theta)]` `=tan^(-1)[(sqrt(sec^(2)theta)-1)/(tan theta)]` `=tan^(-1)[(sec theta-1)/(tan theta)]=tan^(-1)[((1)/(cos theta)-1)/((sin theta)/(cos theta))]` `=tan^(-1)[(1-cos theta)/(sin theta)]=tan^(-1)[("2 sin"^(2)(theta)/(2))/("2 sin"(theta)/(2),"cos"(theta)/(2))]" "({:(because 1-cos theta = "2 sin"^(2)(theta)/(2)and),(" "sin x = "2 sin"(x)/(2)."cos"(x)/(2)):})` `=tan^(-1)["tan"(theta)/(2)]` `rArr y = (theta)/(2) rArr y = (tan^(-1)x)/(2)" "[because theta = tan^(-1)x]` `rArr y = (u)/(2)` `(dy)/(du)=(1)/(2)` `therefore` Option (b) is correct.