Notations for differentiation

The simplest notation for differentiation that is in current use is due to Lagrange and uses the prime, ′. To take derivatives of f(x) at the point a, we write:

f ′(a) for the first derivative,

f ″(a) for the second derivative,

f ″′(a) for the third derivative and then

f⁽ⁿ⁾(a) for the nth derivative (n > 3).

For the function whose value at each x is the derivative of f(x), we write f ′(x). Similarly, for the second derivative of f we write f ″(x), and so on.

The other common notation for differentiation is due to Leibniz. For the function whose value at x is the derivative of f at x, we write:

We can write the derivative of f at the point a in two different ways:

If the output of f(x) is another variable, for example, if y=f(x), we can write the derivative as:

Higher derivatives are expressed as

or

for the n-th derivative of f(x) or y respectively. Historically, this came from the fact that, for example, the 3rd derivative is:

which we can loosely write as:

Dropping brackets gives the notation above.

Leibniz's notation is versatile in that it allows one to specify the variable for differentiation (in the denominator). This is especially relevant for partial differentiation. It also makes the chain rule easy to remember, because the "d" terms appear symbolically to cancel:

However, it is important to remember that the "d" terms cannot literally cancel, because on their own they are undefined. They are only defined when used together to express a derivative.

Newton's notation for differentiation was to place a dot over the function name:

and so on.

Newton's notation is mainly used in mechanics, normally for time derivatives such as velocity and acceleration, and in ODE theory. It is usually only used for first and second derivatives.