Pre amble:
Suppose you have ln(z)=sln(z)=s. This may be rewritten as z=e^sz=es.
You also need to know that if we have d/(ds)(e^s)dds(es) then the solution is d/(ds)(e^s) = e^sdds(es)=es
Back to the question:
y=ln(sqrt(x)) = ln(x^(1/2)) y=ln(√x)=ln(x12)
So x^(1/2) = e^yx12=ey or alternatively squaring both sides gives:
x=(e^y)^2x=(ey)2 ...................................(1)
The differential of this is:
(dx)/(dy) = 2(e^y)dxdy=2(ey) ...............................(2)
But we need (dy/dx)(dydx) this can be achieved by inverting (2) giving
(dy)/(dx) = [2(e^y)]^(-1) = 1/(2(e^y))dydx=[2(ey)]−1=12(ey)................(3)
But from (1) we have e^y = xey=x so by substation in (3) we have:
(dy/dx)=1/(2x)(dydx)=12x
