The retaionship between a magnitude of a funtion and it's rate of change is fairly straightforward. a functions rate of change is simply its derivative. Let's do a few examples.

To find the rate of change of this function, simply find its derivative (as stated above).

Notice the rate of change of this function is constant. Lets do a slightly more complex example.

Notice how with each increased unit for x, the rate of changed is doubled. Lets expand this concept a little farther.

For this function, the rate of change increases even faster with each progressive x value. As a rule, the higher the magnitude of a function, the greater the rate of change for that function. It really is that simple!

source: Calculus with Analytic Geometry: Eighth Edition. Houghton Mifflin Company, Boston. pages 96-114 (first example taken from page 98)

To find the rate of change of this function, simply find its derivative (as stated above).

Notice the rate of change of this function is constant. Lets do a slightly more complex example.

Notice how with each increased unit for x, the rate of changed is doubled. Lets expand this concept a little farther.

For this function, the rate of change increases even faster with each progressive x value. As a rule, the higher the magnitude of a function, the greater the rate of change for that function. It really is that simple!

source: Calculus with Analytic Geometry: Eighth Edition. Houghton Mifflin Company, Boston. pages 96-114 (first example taken from page 98)