Using limits to find the derivative

I'm reading through and can't seem to figure out how to use the limit process here:

f(x)= 4 / x^(1/2)

lim (Δx -> 0) (f(x + Δx) - f(x)) / Δx

???

f(x) = 4 / x^(1/2)
f(x + h) = 4 / (x + h)^(1/2)

(f(x + h) - f(x)) / (h) =>
(4 / (x + h)^(1/2) - 4 / x^(1/2)) / h =>
4 * (1 / (x + h)^(1/2) - 1 / x^(1/2)) / h
4 * ((x^(1/2) - (x + h)^(1/2)) / (x * (x + h))^(1/2)) / h =>
4 * (x^(1/2) - (x + h)^(1/2)) / (h * (x * (x + h))^(1/2))

Rationalize the numerator by multiplying the numerator and denominator by x^(1/2) + (x + h)^(1/2)

4 * (x - (x + h)) / (h * (x * (x + h))^(1/2) * (x^(1/2) + (x + h)^(1/2))) =>
4 * (x - x - h) / (h * (x^2 + hx)^(1/2) * (x^(1/2) + (x + h)^(1/2)))
-4h / (h * (x^2 + hx)^(1/2) * (x^(1/2) + (x + h)^(1/2))) =>
-4 / ((x^2 + hx)^(1/2) * (x^(1/2) + (x + h)^(1/2)))

Let h go to 0

-4 / ((x^2 + 0)^(1/2) * (x^(1/2) + (x + 0)^(1/2)))
-4 / ((x^2)^(1/2) * (x^(1/2) + x^(1/2)))
-4 / (x * 2 * x^(1/2)) =>
-2 / x^(3/2)