Mechanics 1: Linear Kinematics and Calculus - 3D Math Primer for Graphics and Game Development - page 498

11.4.3 Calculating Derivatives from the Definition
Now we're ready for the o cial 20 definition of the derivative found in most
math textbooks, and to see how we can compute derivatives using the
definition. A derivative can be understood as the limiting value of ∆x/∆t,
the ratio of the change in output divided by the change in input, taken
as we make ∆t infinitesimally small. Let's repeat this description using
mathematical notation. It's an equation we gave earlier in the chapter,
only this time we put a big box around it, because that's what math topics
do to equations that are definitions.

The Definition of a Derivative

dx
dt = lim

∆x
∆t = lim

x(t + ∆t) − x(t)
∆t

. (11.3)

∆t→0 ∆t→0

Here the notation for the derivative dx/dt is known as Leibniz's notation.
The symbols dx and dt are known as infinitesimals. Unlike ∆x and ∆t,
which are variables representing finite changes in value, dx and dt are sym-
bols representing “an infinitesimally small change.” Why is it so important
that we use a very small change? Why can't we just take the ratio ∆x/∆t
directly? Because the rate of change is varying continuously. Even within
a very small interval of ∆t = .0001, it is not constant. This is why a limit
argument is used, to make the interval as small as we can possibly make
it—infinitesimally small.
In certain circumstances, infinitesimals may be manipulated like alge-
braic variables (and you can also attach units of measurement to them and
carry out dimensional analysis to check your work). The fact that such
manipulations are often correct is what gives Leibniz notation its intuitive
appeal. However, because they are infinitely small values, they require spe-
cial handling, similar to the symbol ∞, and so should not be tossed around
willy-nilly. For the most part, we interpret the notation d dt not as a ratio of
two variables, but as a single symbol that means “the derivative of x with
respect to t.” This is the safest procedure and avoids any chance of the
aforementioned willy-nilliness. We have more to say later on Leibniz and
other notations, but first, let's finally calculate a derivative and answer the
burning question: how fast was the hare traveling at t = 2.5?

20 We use the word “o cial” here because there are other ways to define the derivative
that lead to improved methods for approximating derivatives numerically with a com-
puter. Such methods are useful when an analytical solution is too di cult or slow to
compute.