Graph of f '(x)

Ever think of drawing the graph of y = f '(x)? First, let's take a look at the graph below. The blue solid curve is the graph of f (x), and the green dotted curve is f '(x).



Recall the concept f '(a) is the slope of f(x) at x = a, while comparing with the graph above:
a) if f(x) is increasing, f '(x) > 0, which means the graph of y = f '(x) falls above the x-axis;
b) if f(x) is constant or reach to an extremum, f '(x) = 0, which means the graph of y = f '(x) falls on the x-axis;
c) if f(x) is decreasing, f '(x) < 0, which means the graph of y = f '(x) falls below the x-axis.

If you wish try out your understanding towards the concept above, perhaps you can ask your friend's help to give you a graph of any function (remember, a function is a one-to-one or many-to-one relationship) and you may try to sketch its derivative graph.

Betty, Created with GeoGebra

Secant & Tangent

The graph below shows the process of getting instantaneous rate of change (slope of tangent) from average rate of change (slope of secant).

The red curve is given as y = g(x). Take note that the average rate of change of y from point A to point B is
.
Drag point B to observe the change of the slope of secant line, m_s. As point B is getting nearer to point A, the value of the slope of secant line is approaching to the value of the slope of tangent line at point A, m_t.



Betty, Created with GeoGebra