Calculus is often used as the complete name of a branch of mathematics dealing with rates of change and very small numbers. In fact, a calculus is a specific discipline or method for the analysis of some set of problems. The fact that the calculus of infinitesimals has come to be known in this way is a testament to its importance.

To understand calculus, it is critical to understand the derivative. Let us begin with the definition of a function.

This expression defines a relation between two quantities x and y. It can be said that y is given in terms of x. Specifically, we have defined a function f that provides a mapping for values of x. The actual mapping of a given function varies; what follows is one example.

Here, the value of the function is said to be equal to the product of its argument, and a constant a. Take note of the fact that this is a constant multiple, as it means that the function is said to be linear--an important property. Evidently, by the transitive property. Here, the relation between y and x becomes clear; y is greater than x by a multiple a. Observe the pattern that forms when evaluating f at various x.