So, it's assuming you already have some informal notion of growth rate in your head, like being able to talk about the velocity of an object even when that velocity is not constant. (Imagine the x coordinate is time, and the y coordinate is position (we'll work in one spatial dimension here); then the "growth rate" is velocity.) Then it discusses how to define this formally.
Unfortunately, I don't have an informal notion of growth rate in my head :/
No. We're talking about instantaneous velocity. You know, the thing the speedometer displays. How fast is the car moving at any given moment? Like, a car doesn't need to be moving at constant speed for a speedometer to give meaningful information, right? Sometimes it is moving faster and sometimes it is moving slower. Sometimes it is moving at a rate such that if it stayed at that rate it would go 60 miles in an hour, and sometimes it is moving at a rate such that if it stayed at that rate it would go 30 miles in an hour. This is the informal notion of instantaneous velocity you should already have. Now the question becomes, how do we formalize this? Which is what the page is trying to answer.
The growth over any finite window, if you partition it into smaller windows, is the sum of the growth within each partition.
Draw enough pictures and you'll develop the intuition that, if you keep partitioning smaller and smaller, you'll reach a point where the average growth rate across a partition is never going to change very much by subpartitioning further. If instantaneous growth rate is going to be defined at all, it has to be very close to the average rate over that tiny interval, no?
Common person or engineer only needs to accept/trust that point in a curve has well defined 'slope' and it's not an approximation.
Actually, what does growth rate even mean?
Think of your morning drive to work. Your distance from the office is always changing as you accelerate and decelerate along the way. It may even stop changing at various times (when you're parked, at stop signs or red lights). If you were to record your distance over time, it would be a curve that goes down (and up when you go in reverse), and sometimes remains level.
The slope of that curve at a given point, that is the derivative, corresponds to the speed displayed by your speedometer at that particular moment in time.
This is differential calculus in a nutshell: given a curve representing your distance over the entire trip, find the speed at any instant in time. We can then relate this to integral calculus in the following way: given a way to record your speed at every moment in time (speedometer), determine the total distance you travel. If it sounds like two sides of the same coin, well it is! This is the brilliant discovery of the fundamental theorem of calculus.
