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