Given that, #y=sqrtx+1/sqrtx=x^(1/2)+x^(-1/2)#.
Recall that, #d/dx(x^n)=n*x^(n-1)#.
#:. dy/dx=1/2*x^(1/2-1)+(-1/2)*x^(-1/2-1)#.
# =1/2*x^(-1/2)-1/2*x^(-3/2)#,
#:. dy/dx=1/2{x^(-1/2)-x^(-3/2)}#.
Multiplying this eqn. by #2x#, we get,
#2xdy/dx=cancel(2)x*1/cancel(2){x^(-1/2)-x^(-3/2)}#,
#=x*x^(-1/2)-x*x^(-3/2)#,
# i.e., 2xdy/dx=x^(1/2)-x^(-1/2)=sqrtx-1/sqrtx#.
Finally, adding #y=sqrtx+1/sqrtx#, we have,
#2xdy/dx+y=(sqrtxcancel(-1/sqrtx))+(sqrtxcancel(+1/sqrtx))#,
# or, 2xdy/dx+y=2sqrtx#,
as Respected Abhishek Malviya has readily derived!
