The Notation of Differentiation
Suggested Prerequesites:
The definition of the derivative
Often the most confusing thing for a student introduced to
differentiation is the notation associated with it. Here I will attempt
to introduce as many types of notation as possible.
derivative is always the derivative of a function with respect
to a variable. When we write the definition of the derivative as
we mean the derivative of the function f(x) with respect to
the variable x.
One type of notation for derivatives is sometimes called prime
notation. f'(x), which would be read "f
prime of x," means the derivative of f(x) with
respect to x. If we say y = f(x), then
y' (read "y prime") = f'(x).
This is even sometimes taken as far as to write things such as, for
y = x4 + 3x (for example), y' =
(x4 + 3x)'
Higher order derivatives in prime notation are
represented by increasing the number of primes. For example, the second
derivative of y with respect to x would be written
as y''. Beyond the second or third derivative, all those
primes get messy, so often the order of the derivative is instead writen
as a superscript in parenthesis, so that the ninth derivative of
f(x) with respect to x is written as
f(9)(x).
A second type of notation for derivatives is sometimes called
operator notation. The operator Dx
is applied to a function in order to perform differentiation. Then, the
derivative of f(x) = y with respect to x can be
written as Dxy (read "D sub x of
y") or as Dxf(x) (read "D sub
x of f(x)").
Higher order derivatives are written by adding a superscript to
Dx, so that the third derivative of y =
(x2 + sin x)
with respect to x would be written as
Dx3y =
Dx3(x2 + sin
x).
Another commonly used notation was developed by
Leibnitz and is accordingly called Leibnitz notation. With
this notation, if y = f(x), then the derivative of
y with respect to x can be written as
. This is read as "d y d x" or
sometimes "d y over d x." Since y = f(x), we can also write
. This notation suggests that
perhaps derivatives can be treated like fractions, which is true in
limited ways in some circumstances. (For example with the
chain rule.) This is also called differential
notation, where dy and dx are
differentials. This notation becomes very useful when dealing
with differential equations.
A variation of Leibnitz's differential notation is written instead as
, which resembles the above operator
notation, with
as the operator.
Higher order derivatives using leibnitz notation can be written as
or as
.
The exponents may seem to be in strange places in the second form, but
it makes a wierd sort of sense if you look at the first form.
So, those are the most commonly used notations for differentiation. It's
possible that there exist other, obscure notations used by a few people,
but I've never heard of any others. It's helpful to be familiar with
the different notations.
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jjnichol@mit.edu
Last modified 23 June 1997