 Let's recap the essential concepts about the derivative of a function at a point. First, we begin by realizing that the concept of average velocity can be generalized to talk about the rate of change in any function. We're going to define for any function f the average rate of change of f on the interval from a to a plus h, to be given by the formula f of a plus h minus f of a divided by h. Or if we choose to think of the interval as going from a to another point b, then the average rate of change of f on ab is f of b minus f of a over b minus a. Both of these are just the average velocity formulas from earlier in the course, except now we're thinking of f as being any function, not necessarily a function that gives us position at a certain time. Next, we introduce the central idea of this course, and that is the notion of the derivative of a function at a point. We've previously defined the instantaneous velocity of a moving object to be its velocity right at a single point in time. We calculated this instantaneous velocity by setting up the average velocity formula on the interval from a to a plus h, where h is some small separation in time. We calculated that average velocity with an h in it, and then we took the limit as h goes to zero. This shrinks the length of the time interval down to basically nothing, and instead of giving us an average velocity, it gives us a velocity at a single moment. So we just adapt that formula here to define the derivative of a function f at a point to be f prime of a, and that is calculated as the limit as h goes to zero of f of a plus h minus f of a divided by h. Notice that the fraction here is just the average velocity formula, and the limit is indicating that we are shrinking the time interval down. Finally, we know that there are three co-equal ways of interpreting what the derivative of a function tells us at a certain point. The derivative of a function at a point tells us, first of all, the slope of the tangent line to the graph of f at that point. Secondly, it tells us the instantaneous velocity of a moving object, whose position at a given time t is given by f. And finally, it tells us the rate at which f is changing exactly at the moment that x equals a. That is also known as the instantaneous rate of change.