Intuitive definition of "rate of attraction" of a fixed point

Question

If possible, I would like to know if there is an easy and intuitive definition of the "rate of attraction" of a fixed point. I am especially interested in the difference between super-attractive and simple-attractive fixed point (e.g.: $x = \sqrt{\alpha}$ is a super-attractive fixed point of $f(x) = \frac{1}{2}(x+\frac{\alpha}{x})$, whereas it is a simple-attractive fixed point of $g(x) = \frac{x + \alpha}{x + 1}$).