The Average Gradient of a Function Between Two Points

Suppose that a function f is defined on some open interval (a,b) on the x-axis.
Then we choose a fixed point x in this interval and introduce the difference quotient
or the average gradient. of the function.
where the number h may be positive or negative (but not zero) and is such that x+h also lies in (a,b).
The numerator of this quotient measures the change in the function when x changes from x to x+h.
The quotient itself is often referred to as the average rate of change of f in the interval joining x to x+h.