For a function 'f' of an argument 'x', the derivative operator is sometimes given:

For the derivative operator , an eigenfunction is a function that, when differentiated, yields a constant times the original function.

Since the law applies only to systems of constant mass, m can be brought out of the derivative operator.

Note that although the derivative operator is not continuous, it is closed.

Newton's second law applies only to a system of constant mass, and hence m may be moved outside the derivative operator.

These are abbreviations for multiple applications of the derivative operator.

If, however, one defines carefully the domain and range of the derivative operator, one may show that it is a closed operator.

At any scale in scale space, we can apply local derivative operators to the scale-space representation:

The derivative operator is defined also for complex-valued functions of a complex argument.

A simple example of an infinite matrix is the matrix representing the derivative operator, which acts on the Taylor series of a function.