Mechanics 1: Linear Kinematics and Calculus - 3D Math Primer for Graphics and Game Development - page 495

the derivative of that function at t is the ratio dx/dt. The symbol dx rep-
resents the change in the output produced by a very small change in the
input, represented by dt. We'll speak more about these “small changes” in
more detail in just a moment.
For now, we are in an imaginary racetrack where rabbits and turtles race
and moral lessons are taught through metaphor. We have a function with
an input of t, the number of minutes elapsed since the start of the race, and
an output of x, the distance of the hare along the racetrack. The rule we use
to evaluate our function is the expression x(t) = −t 2 +6t−1. The derivative
of this function tells us the rate of change of the hare's position with respect
to time and is the definition of instantaneous velocity. Just previously, we
defined instantaneous velocity as the average velocity taken over smaller
and smaller intervals, but this is essentially the same as the definition of
the derivative. We just phrased it the first time using terminology specific
to position and velocity.
When we calculate a derivative, we won't end up with a single number.
Expecting the answer to “What is the velocity of the hare?” to be a single
number makes sense only if the velocity is the same everywhere. In such
a trivial case we don't need derivatives, we can just use average velocity.
The interesting situation occurs when the velocity varies over time. When
we calculate the derivative of a position function in such cases, we get a
velocity function, which allows us to calculate the instantaneous velocity at
any point in time.
The previous three paragraphs express the most important concepts in
this section, so please allow us to repeat them.
A derivative measures a rate of change. Since velocity is the rate of change
of position with respect to time, the derivative of the position function is
the velocity function.
The next few sections discuss the mathematics of derivatives in a bit
more detail, and we return to kinematics in Section 11.5. This material is
aimed at those who have not had 14 first-year calculus. If you already have
a calculus background, you can safely skip ahead to Section 11.5 unless you
feel in need of a refresher.
Section 11.4.2 lists several examples of derivatives to give you a better
understanding of what it means to measure a rate of change, and also to
14 The authors define “had” to mean that you passed it, you understood it, and you
remember it.