Part 2/6:
First Derivative and Its Implications
The first derivative ( f'(x) ) indicates the slope or rate of change of the function ( f(x) ). When ( f'(x) > 0 ), the function is increasing. As we examine various positions on the graph, we notice differences in how steeply the function rises or falls.
For instance, suppose we have two functions that are both increasing; one may increase more sharply than the other. This is determined by investigating the behavior of the first derivative across different intervals.
Second Derivative and Concavity
The second derivative ( f''(x) ) provides insight into the rate of change of the first derivative, thereby indicating the concavity of the graph.