Increments

If  is any function, the expression  represents the change of y (increment of y) or change of f from point  to point . This change is denoted by  (pronounced delta y) or .

The expression  represents the increment of x (change in x from point  to point ). It is denoted by .

Dividing the change of y by the change of x, we get the average rate of change of y with respect to x from point  to p. That average rate of change equals:

To get the instantaneous rate of change of the function, we will have to take the limit of this quotient as  approaches 0. That is,

We call this limit the first derivative of the function f. We denote it by , ,  and . Note well that the first derivative is also a function of x.