SECTION 2.2: The Derivative Function

The derivative of a function can be estimated graphically by finding the slope of the tangent to a point on a graph. This is so because, in general, the derivative function f '(x) is defined as the instantaneous rate of change of f at x.

Graphically, the derivative graph tells us that,
if f ' > 0, then f is increasing
if f ' < 0, then f is decreasing and
if f ' = 0, then f is constant.

Note: If the magnitude of f ' is large then the graph of the derivative is steep. Conversely, if the magnitude if small, the graph is gently sloping.

A derivative can also be obtained/ estimated from a table of data by finding the change in the x values of the data and dividing it by the change in y values of the data. Essentially, this implies finding the slope of the data when graphed.

Derivatives of certain points can also be numerically estimated by studying the average of rates of changes of point to the left and to the right of it.

What I found confusing in this reading, was the suggestion that derivatives of points could also be numerically calculated by just analyzing points to the right of it.

This reading was very useful in economics as learning derivatives in general helps understand market situations relating to recessionary, inflationary or other phases of the business cycle. For example when f ' = 0, it would imply that the market is going through a lull/ no activity phase.