Summary

(1) The average rate of change of a function over a certain interval is
the same as the gradient of the chord drawn on the graph of the
function. This chord gradient, for a function, y = f (x), is given by

(2) If the chord is very short then the gradient of the chord is approximately
the gradient of the tangent to the graph at a particular spot that is, the slope of the graph at that point. The slope of the graph
gives the instantaneous rate of change of the function with respect to
its independent variable, known as its derivative. This is represented
by dy/dx. Then we have the definition