Derivatives as an Instantaneous Rate of Change
Here we'll approach the derivative from a different and more real-lifey direction.
There are two ways to think about speed while driving. The first way is to look at the speedometer, which shows how fast the vehicle is going at that moment. The other way is to take the number of miles driven and divide by how long the drive took. The first way tells the "instantaneous" speed—the speed at that instant. The second way gives average speed over the whole trip.
Although looking at the speedometer may seem like the best way to figure out speed, in order to relate this driving stuff properly to derivatives we need to talk about average speed first.