So what is the derivative, after all?
UBC Calculus Online Course Notes

Summary and suggestions for further study

In this unit, we have seen the derivative appear in several guises.

  1. As the slope of a graph when viewed under a very powerful microscope.
  2. As the instantaneous rate of change of some physical process.
  3. If a function repents the position of an object as a function of time, then the function's derivative represents the instantaneous velocity of the object.

It can be confusing to see the same idea from so many different angles (remember the blind men touching the elephant?). If so, don't worry too much because the rest of the course is intended to explore these ideas more deeply. It will help you to see, as we will shortly, what the derivative is used for. Gradually, you will be able to move back and forth between these seemingly different interpretations with ease. What you should recognize in all three scenarios is that the derivative is measuring how a function changes over a very small scale (infinitesimally small, in fact).

You might want to explore a bit further and see what other folks in physics and mathematics courses have to say about this and related examples. The links below will help you get started. You might also want to search the web for the topics "projectile motion", "falling ball" and related concepts that we have discussed here.

Other sites to visit
Note: The falling Ball is an example of what is called 1 dimensional motion in mechanics. It is related to more complicated motions (such as the motion of a thrown ball in 2 and 3 dimensions). We will not discuss these more complicated motions now, but you may want to peak at what other people have to say about these by visiting: