12.1 Differentiation in theory
Instead of jumping straight into the mathematical definition, let’s start our discussion with a straightforward example: a point-like object moving along a straight line. Its movement is fully described by the time-distance plot (Figure 12.1), which shows its distance from the starting point at a given time.
Our goal is to calculate the object’s velocity at a given time. In high school, we learned that
To put this into a quantitative form, if f(t) denotes the time-distance function, and t1/span>t2 are two arbitrary points in time, then
Expressions like 
are called differential quotients. Note that if the object moves backwards, the average velocity is negative. (As opposed to speed, which is always positive. Velocity is speed and direction.)
The average velocity has a simple geometric interpretation: if you replace the object’s motion with a constant velocity...
