It will be presumed here that you meant exactly what you said, and want the differential #dy# rather than the derivative #dy/(dx)#. If you instead want the derivative, simply ignore the last step.
Recall the power rule states that for any term #ax^b#, with a and b as constants, the derivative is #dy/(dx) = (b)ax^(b-1)#. Recall further that #sqrtx = x^(1/2)# and that #1/sqrtx = 1/x^(1/2) = x^(-1/2)# (because any term #p/x^q = px^(-q)# by the properties of exponents). Thus, if given the function above, we can find its derivative (and differential) as follows:
#y =sqrtx + 1/sqrtx= x^(1/2)+x^(-1/2)#.
#-> dy/(dx) = d/(dx)x^(1/2)+ d/(dx)x^(-1/2) = 1/2x^(-1/2) - 1/2x^(-3/2)#
#dy/(dx)= 1/2x^(-1/2) - 1/2x^(-3/2)#
This, we now have the derivative. If we want the differential #dy#...
#dy/(dx)= 1/2x^(-1/2) - 1/2x^(-3/2) -> dy = ( 1/2x^(-1/2) - 1/2x^(-3/2))dx#
