Use the given graph to investigate the second derivative (i.e., the derivative of the derivative). [br][list][*]Use the "Tangent on f" checkbox to show/hide a tangent line segment on f at P. This will also show a point on the right graph that represents the slope of the tangent as a y-coordinate. Moving P will leave a trace of this slope at each point to trace the graph of [math]f'(x)[/math]. [/*][*]The checkbox for [math]f'(x)[/math] will show the graph of the derivative on the left. The "Tangent on f'" checkbox will make a point Q appear on the graph of f' with its own tangent line segment. Again, a point (in orange) will appear on the right to represent the slope of this tangent line as a y-coordinate. Move Q along the graph to trace the graph of the second derivative function. [/*][*]Use the [math]f''(x)[/math] checkbox to show/hide the graph of the second derivative function. [/*][/list]

Because the derivative of a function is itself a function (with its own formula, table, graph, etc.), it also has its own rate of change at each point. [i]Thinking in terms of motion[/i], the derivative of position is velocity. At any given time a moving object has a velocity, but we know from experience that the velocity is not necessarily constant. [i]The velocity, which is a rate of change, has its own rate of change, which we call acceleration.[/i] [br][br]Position, velocity, and acceleration are physical examples of the more abstract mathematical concepts of a function, its (first) derivative, and its [b]second derivative[/b]. In other words, [i]the second derivative of a function is the derivative of its derivative, [/i]denoted by [math]f''[/math][i].[/i] You may be able to anticipate how we could continue this process and talk about third derivatives, fourth derivatives, etc., but the physical interpretations become less meaningful. [br][br]Reflection: We know that (first) derivatives give information about whether a function is increasing or decreasing. What can we say about f' when f'' is positive? What does that tell us about f? (We will explore these types of relationships in more detail later.)