Question 1164542
In order to compute the derivative from first principles, we need to determine the following:
lim(h->0)[(sqrt(x+h) - sqrt(x))/h]
Rationalize the numerator by multiplying top and bottom by sqrt(x+h) + sqrt(x).  
This gives lim(h->0)[(x+h - x)/(h*(sqrt(x+h) + sqrt(x)))]
This reduces to lim(h->0)[1/(sqrt(x+h) + sqrt(x))]
Letting h go to zero, we get 1/(2*sqrt(x))