Differentiate each of the following w.r.t. x:
`(i) tan^(-1)(sqrt(1+x^(2))+x)" "(ii)tan^(-1)(sqrt(1+x^(2))-x)`

Explanation

(i) Let `y=tan^(-1)(sqrt(1+x^(2))+x).`
Putting `c=cot theta,` we get
`y=tan^(-1)("cosec "theta+cot theta)=tan^(-1)((1)/(sin theta)+(cos theta)/(sin theta))`
`=tan^(-1)((1+cos theta)/(sin theta))=tan^(-1)((2cos^(2)(theta//2))/(2sin(theta//2)cos (theta//2)))`
`=tan^(-1)("cot"(theta)/(2))=tan^(-1){tan((pi)/(2)-(theta)/(2))}`
`=(pi)/(2)-(1)/(2)theta=(pi)/(2)-(1)/(2)cot^(-1)x.`
`therefore (dy)/(dx)=-(1)/(2).(-1)/((1+x^(2)))=(1)/(2(1+x^(2))).`
Hence, `(d)/(dx){tan^(-1)(sqrt(1+x^(2))+x)}=(1)/(2(1+x^(2))).`
Let `y=tan^(-1)(sqrt(1+x^(2))-x)`
Putting `x=cot theta`, we get
`y=tan^(-1)("cosec "theta-cot theta)=tan^(-1)((1)/(sin theta)-(cos theta)/(sin theta))`
`=tan^(-1)((1-cos theta)/(sin theta))=tan^(-1){(2sin^(2)(theta//2))/(2sin(theta//2)cos(theta//2))}`
`=tan^(-1)(tan.(theta)/(2))=(1)/(2)theta=(1)/(2)cot^(-1)x.`
`therefore(dy)/(dx)=(-1)/(2(1+x^(2))).`
Hence, `(d)/(dx){tan^(-1)(sqrt(1+x^(2))-x)}=(-1)/(2(1+x^(2))).`