Average rate of change = (change in y)/(change in t)
Instantaneous rates of change
Of course, the value of the average rate that we computed depends not only on the point identified as P, but also on the second point Q on the graph. If we were to chose a different endpoint for the same calculation, i.e. a different point for Q, we would get a different average rate of change.
In the next picture, we show what happens when we make the second point closer and closer to P. Notice that, as Q is chosen closer to P, the secant lines (shown in blue) have slopes that measure the change that take place very close to P.
Notice that the sequence of secant lines shown in the previous picture accumulate around a unique line through the point P. That line is called the tangent line. It has been drawn here in red, together with the secant lines, to show their relationship. The tangent line represents a limiting process in which the average rate of change is calculated over smaller intervals around P. As before, we say that this function is differentiable at P, and we call the slope of the tangent line the derivative at P. Since the derivative is obtained by measuring the average rate of
change close to P, we can think of it as measuring an instantaneous rate of change.
Here is a way to see this in an interactive fashion. Simply click somewhere on the graph. You will then see the secant line which you can then drag towards the basepoint. As you approach the basepoint, the secant line approaches the tangent line which is shown in red.
There is nothing special about the point P in our example, and we could as easily have considered any point on the curve as a location at which the tangent line is to be found. In the graph below, you can see the tangent line drawn at several different points along the curve. The slopes of these tangents change from point to point. As we will later discuss, the behaviour of these slopes are in themselves an interesting trend (and will form a function of time that will be called the derivative of the original function.)
Here we see that the secant lines on the right all have slope 1 while the secant lines on the left all have slope -1. This means that the secants do not approach a unique line and so we say that the absolute value function is not differentiable at x = 0.