Often the most confusing thing for a student introduced to differentiation is the notation associated with it. Here an attempt will be made to introduce as many types of notation as possible.

A 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**. The function `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` = `x`^{4} + 3x
(for example), `y`´ =
(`x`^{4} + 3`x`)´.

**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

Beyond the second or third derivative, all those primes get messy, so
often the order of the derivative is instead writen as a roman
superscript in parenthesis, so that the ninth derivative of
`f`(`x`) with respect to `x` is written as
`f`^{(9)}(`x`) or
`f`^{(ix)}(`x`).

A second type of notation for derivatives is sometimes called

Higher order derivatives are written by adding a superscript to

Another commonly used notation was developed by Leibnitz and is accordingly called

(his is read as ``dy -- dx'', but not ``dy minus dx'' or
sometimes ``dy over dx''). 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 (`d`/`dx` as the operator).

Higher order derivatives using leibnitz notation can be written as

The exponents may seem to be in strange places in the second form, but it makes 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 some, but these obscure forms won't be included here. It's helpful to be familiar with the different notations.

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watko@mit.edu Last updated August 24, 1998 <\body>