Find the derivative by rule:
dy/dx 1/2√x
Did you mean "by first principles" ?
let f(x) = (1/2)√x
then f(x+h) = (1/2)√(x+h)
dy/dx = Lim ((1/2)√x - (1/2)√(x+h) )/h, as h ---> 0
= lim (1/2)(√(x+h) - √x)/h * (√(x+h) + √x)/(√(x+h) + √x) , as h ---> 0
= lim (1/2) (x+h - x)/(h((√(x+h) + √x) )
= lim (1/2) h/(h((√(x+h) + √x) ) , as h ---> 0
= (1/2) 1/(2√x)
= 1 / (4√x)