Differentiate `sqrt(cot^(-1)sqrtx)`, w.r.t. x.

Explanation

Let `y=sqrt(cot^(-1)sqrtx)`.
Putting `sqrtx=t and cot^(-1)sqrtx=cot^(-1)sqrtx=u`, we get
`y=sqrtu,` where `u=cot^(-1)t and t=sqrtx`.
Now, `y=sqrtu rArr (dy)/(du)=(1)/(2)u^(-1//2)=(1)/(2sqrtu),`
`u=cot^(-1)t rArr (du)/(dt)=(-1)/((1+t^(2)))`.
And, `t=sqrtx rArr (dt)/(dx)=(1)/(2)x^(-1//2)=(1)/(2sqrtx)`.
`therefore(dy)/(dx)=((dy)/(du)xx(du)/(dt)xx(dt)/(dx))=(-1)/(4sqrtu(1+t^(2))sqrtx)`
`" "=(-1)/(4(sqrt(cot^(-1)t))(1+t^(2))sqrtx)" "[because u=cot^(-1)t]`
`" "=(-1)/(4(sqrt(cot^(-1)sqrtx))(1+x)sqrtx)" "[because t=sqrtx].`