Let f(x)=sqrt(x), then
f'(x)=lim_(hto0)(sqrt(x+h)-sqrt(x))/(h)=lim_(hto0)(sqrt(x+h)-sqrt(x))/(h)*(sqrt(x+h)+sqrt(x))/(sqrt(x+h)+sqrt(x))
=lim_(xto0)(x+h-x)/(h(sqrt(x+h)+sqrt(x)))
=lim_(hto0)(h)/(h(sqrt(x+h)+sqrt(x)))
=lim_(hto0)(1)/((sqrt(x+h)+sqrt(x)))
=(1)/(2sqrt(x))
In general, you can use the fact that if f(x)=x^t, then f'(x)=tx^(t-1)
Taking t=1/2, gives us that f'(x)=(1)/(2)x^(-(1)/(2)), which is the same as we obtained above.
Also, recall that (d(cf(x)))/(dx)=c(df(x))/(dx). Hence, you can pull out the constant and then differentiate it.

The Power Rule says that (d)/(dx)x^(alpha)=alpha x^(alpha-1). Applying this to sqrt(x)=x^((1)/(2)) gives (d)/(dx)sqrt(x)=(d)/(dx)x^((1)/(2)) =(1)/(2)x^(-(1)/(2)) =(1)/(2sqrt(x)) However, if you are uncomfortable applying the Power Rule to a tional power, consider applying implicit differentiation to y=sqrt(x) y^2=x 2y(dy)/(dx)=1 (dy)/(dx)=(1)/(2y) =(1)/(2sqrt(x))