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We do have room, however, to mention alternate notations for the
derivative that you are likely to encounter.
11.4.4 Notations for the Derivative
Several different notations for derivatives are in common use. Let's point
out some ways that other texts might look different from what we've said
here. First of all, there is a trivial issue of naming. Most calculus textbooks
define the derivative in very general terms, where the output variable is
named y, the symbol x refers to the input variable rather than the output
variable, and the function is simply named f. In other words, the function
being differentiated is y = f(x). Furthermore, many will assign the shrink-
ing “step amount” to the variable h rather than using the ∆ notation, which
has advantages when solving the equations that result when you work out
derivatives from the definition.
22
With these variables, they would define
the derivative as
Definition of a derivative
using variables in most
calculus textbooks
dy
dx
= lim
y(x + h) − y(x)
h
.
(11.6)
h→0
The differences between Equations (11.3) and (11.6) are clearly cosmetic.
A variation on the Leibniz notation we prefer in this topic is to prefix
an expression with d/dt to mean “the derivative with respect to t of this
thing on the right.” For example
d
dt
(t
2
+ 5t)
can be read as “the derivative with respect to t of t
2
+ 5t.” This is a very
descriptive and intuitive notation. If we call the expression on the right x,
and interpret the juxtaposition of symbols as multiplication, we can pull
the x back on top of the fraction to get our original notation, as in
dt
(t
2
+ 5t) =
d
d
dt
x =
dx
dt
.
It's important to interpret these manipulations as notational manipu-
lations rather than having any real mathematical meaning. The notation
is attractive because such algebraic manipulations with the infinitesimals
often work out. But we reiterate our warning to avoid attaching much
mathematical meaning to such operations.
Another common notation is to refer to the derivative of a function
f(x) with a prime: f
′
(x). This is known as prime notation or Lagrange's
22
Notice our klunky need for parentheses with (∆t)
2
to avoid the potentially confusing
notation ∆t
2
.
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