Suppose f is a function of x. Consider two points P(a, f(a)) and Q(b, f(b)) on the graph of the function f(x).

The slope of the secant line PQ =

$\frac{f(b)-f(a)}{b-a}$ is called the difference quotient.

The difference quotient of the two points P and Q is defined to be the average rate of change of function from x = a to b.

When the two points are taken as (x, f(x)) and (x + $\Delta$ x, f(x+

$\Delta$ x)), then the average rate of function as x changes from x to x+

$\Delta$x is

Average rate of change =

$\frac{f(x+\Delta x)-f(x)}{\Delta x}$ When
we use the variable y to represent the function, that is when y = f(x)
and if the change in y is Δy corresponding to a change of Δx in x, then the average rate of change of y with respect to x is written as

$\frac{\Delta y}{\Delta x}$$\frac{\Delta y}{\Delta x}$ =

$\frac{f(x+\Delta x)-f(x)}{\Delta x}$