The graph of a function is given on the left, with a movable point and a tangent line segment. The main purpose of this applet is to investigate derivatives graphically for many common functions. So, make sure to change the function f with the input box. Try power functions, exponential functions, trig functions, etc. [br][list][*]A point (green) on the right represents the slope of the tangent line segment as a y-coordinate. Click "Trace Derivative" to leave a trace of this point as you move the point P. This will generate a graph of the derivative function. [/*][*]Click the "[math]f'(x)[/math]" checkbox to show/hide the actual graph of the derivative function as well as the output values in function notation. [/*][/list]

Because the derivative of a function is itself a function, it often has its own formula, graph, and table (as does any function). These different representations of functions are useful for [b]analyzing function behavior [/b](i.e., is it increasing or decreasing? How fast is it changing? Does it have any maximum or minimum values? Is it bending up or bending down? Does that ever change?) We will address all of these questions throughout the course, but it will take some time to develop everything we need. [br][br]As we compare the graph of a function [math]f(x)[/math] and the graph of its derivative [math]f'(x)[/math], we observe a key relationship between the [b]monotonicity [/b](i.e., increasing vs. decreasing) of [math]f[/math] and the [b]sign [/b](i.e., positive or negative) of the derivative [math]f'[/math]. For example, when a function is increasing, its derivative is positive (because the tangent lines have positive slope). Conversely, if we don't know [math]f[/math] but we do know that its derivative is positive, we can say that [math]f[/math] is increasing there. [br][br]Throughout the semester we will see other key relationships between the "behavior" of a function and the "behavior" or properties of its derivative(s).