Mechanics 1: Linear Kinematics and Calculus - 3D Math Primer for Graphics and Game Development - page 504

notation. It's used when the independent variable that we are differen-
tiating with respect to is implied or understood by context. Using this
notation we would define velocity as the derivative of the position function
by v(t) = x ′ (t).
One last notation, which was invented by Newton and is used mostly
when the independent variable is time (such as in the physics equations
Newton invented), is dot notation. A derivative is indicated by putting a
dot over the variable; for example, v(t) = x(t).
Here is a summary of the different notations for the derivative you will
see, using velocity and position as the example:

v(t) = dx

dt = d

′

dt x(t) = x

(t) = x(t).

11.4.5 A Few Differentiation Rules and Shortcuts

Now let's return to calculating derivatives. In practice, it's seldom neces-
sary to go back to the definition of the derivative in order to differentiate
an expression. Instead, there are simplifying rules that allow you to break
down complicated functions into smaller pieces that can then be differen-
tiated. There are also special functions, such as lnx and tanx, for which
the hard work of applying the definition has already been done and written
down in those tables that line the insides of the front and back covers of
calculus topics. To differentiate expressions containing such functions, one
simply refers to the table (although we're going to do just a bit of this
“hard work” ourselves for sine and cosine).
In this topic, our concerns are limited to the derivatives of a very small
set of functions, which luckily can be differentiated with just a few simple
rules. Unfortunately, we don't have the space here to develop the mathe-
matical derivations behind these rules, so we are simply going to accompany
each rule with a brief explanation as to how it is used, and a (mathemati-
cally nonrigorous) intuitive argument to help you convince yourself that it
works.
Our first rule, known as the constant rule, states that the derivative of
a constant function is zero. A constant function is a function that always
produces the same value. For example, x(t) = 3 is a constant function. You
can plug in any value of t, and this function outputs the value 3. Since, the
derivative measures how fast the output of a function changes in response
to changes in the input t, in the case of a constant function, the output
never changes, and so the derivative is x ′ (t) = 0.