Independent and Dependent Variables

Independent and dependent variables are useful in the study of derivatives. Let us briefly review what they are. A system of formulas is a finite set of equations and inequalities. If we are given a system of formulas which has the same graph as a simple equation y = f(x) we say that y is a function of x, or that y depends on x, and we call x the independent variable and y the dependent variable.

When y = f(x), we introduce a new independent variable Δx and a new dependent variable Δy, with the equation

(1)

Δy = f(x + Δx) - f(x).

This equation determines Δy as a real function of the two variables x and Δx, when x and Δx vary over the real numbers. We shall usually want to use the Equation 1 for Δy when x is a real number and Δx is a nonzero infinitesimal. The Transfer Principle implies that Equation 1 also determines Δy as a hyperreal function of two variables when x and Δx are allowed to vary over the hyperreal numbers.

Δy is called the increment of y. Geometrically, the increment Δy is the change in y along the curve corresponding to the change Δx in x. The symbol y' is sometimes used for the derivative, y' = f'(x). Thus the hyperreal equation

now takes the short form

The infinitesimal Δx may be either positive or negative, but not zero. The various possibilities are illustrated in Figure 2.1.2 using an infinitesimal microscope. The signs of Δx and Δy are indicated in the captions.

Our rules for standard parts can be used in many cases to find the derivative of a function. There are two parts to the problem of finding the derivative f' of a function f:

(1)    Find the domain of f'.

(2)    Find the value of f'(x) when it is defined.