The Derivative as a Limit of an Average Rate of Change

In what follows…

We will be introducing the derivative, which can be thought of as an “instantaneous (average) rate of change,” such as being able to determine how fast an object is moving at any given moment. In practice and in the “real world,” determining how fast an object is moving requires measuring the object’s location at two different times (usually taken close together). It is impossible to figure out an object’s speed by taking one measurement of position at only one time. That would be like taking a photo of a car driving on the road, and asking you to determine how fast it’s going based on that photo. You would need two photos of the car moving, and you’d need the time those photos were taken to determine the speed of the car. However, limits and the techniques of calculus essentially give us the ability to theoretically determine the speed of the car based on one single photo (with a timestamp) if we have a function that tells you the car’s position at any given time. This is all made precise in the topics below.