Mechanics 1: Linear Kinematics and Calculus - 3D Math Primer for Graphics and Game Development - page 496

back up our claim that the derivative has very broad applicability. Sec-
tion 11.4.3 gives the formal mathematical definition of the derivative 15 and
shows how to use this definition to solve problems. We also finally figure out
how fast that hare was moving at t = 2.5. Section 11.4.4 lists various com-
monly used alternate notations for derivatives, and finally, Section 11.4.5
lists just enough rules about derivatives to satisfy the very modest differ-
ential calculus demands of this topic.

11.4.2 Examples of Derivatives
Velocity may be the easiest introduction to the derivative, but it is by no
means the only example. Let's look at some more examples to give you an
idea of the wide array of problems to which the derivative is applied.
The simplest types of examples are to consider other quantities that
vary with time. For example, if R(t) is the reading of a rain meter at
a given time t, then the derivative, denoted R ′ (t), describes how hard it
was raining at time t. Perhaps P(t) is the reading of a pressure valve
on a tank containing some type of gas. Assuming the pressure reading
is proportional to the mass of the gas inside the chamber, 16 the rate of

′

change P (t) indicates how fast gas is flowing into or out of the chamber at

time t.
There are also physical examples for which the independent variable is
not time. The prototypical case is a function y(x) that gives the height
of some surface above a reference point at the horizontal position x. For
example, perhaps x is the distance along our metaphorical racetrack and y
measures the height at that point above or below the altitude at the starting
point. The derivative y (x) of this function is the slope of the surface at

x, where positive slopes mean the runners are running uphill, and negative
values indicate a downhill portion of the race. This example is not really a
new example, because we've looked at graphs of functions and considered
how the derivative is a measure of the slope of the graph in 2D.
Now let's become a bit more abstract, but still keep a physical dimension
as the independent variable. Let's say that for a popular rock-climbing
wall, we know a function S(y) that describes, for a given height y, what
percentage of rock climbers are able to reach that height or higher. If we
assume the climbers start at y = 0, then S(0) = 100%. Clearly S(y) is
a nonincreasing function that eventually goes all the way down to 0% at
some maximum height y max that nobody has ever reached.
Now consider the interpretation of derivative S ′ (y). Of course, S ′ (y) ≤
0, since S(y) is nonincreasing. A large negative value of S ′ (y) is an indi-

′

15 Spoiler alert: we already gave it to you in this section!
16 This would be true if we are operating in a range where the ideal gas law is a valid
approximation, and the temperature remains constant.