differentiate:

y=((x^1/2)^(x^1/2))×e^(x^1/2) 

let the integrate be equal to y

y = ((x^1/2)^(x^1/2))×e^(x^1/2)

taking log on  the base e both sides

ln y =ln ((x^1/2)^(x^1/2))  +  x^1/2

       = x^1/2 ln x^1/2 + x^1/2

       = x^1/2   (ln x^1/2   + 1).......................took x^1/2 common

now differentiate by product rule

1/y   (dy/dx)   =  [1/2x^1/2   ]  *  ( ln x^1/2    +  1)        +  x^1/2 *  (1/2x)

dy / dx = ((x^1/2)^(x^1/2))×e^(x^1/2)     *   {[1/2x^1/2   ]  *  ( ln x^1/2    +  1)        +  x^1/2 *  (1/2x)}

ans!!