This applet involves a lot of the same elements as 2-A (The Derivative at a Point), but we will consider specific examples and investigate how the instantaneous rate of change can fail to exist at certain points. [br][br]The graph of a function is shown in the applet. [br][list][*]Use the input box for c or click and drag the point on the graph to change the point where you want to investigate the instantaneous rate of change. [/*][*]Adjust the slider tool for h to move the point Q around P. [/*][*]The "Secant" checkbox will show/hide the secant line between P and Q and the slope of the secant line (i.e., average rate of change). [/*][*]The "Difference Quotient" checkbox will show/hide the graph of the difference quotient function, which has an excluded value when h = 0. [/*][*]The "Tangent" checkbox will show/hide the tangent line at P and its slope. [/*][*]Use the "h \to 0" and observe the relationship between the secant and tangent lines. [/*][/list]

Because the derivative of a function at a point (i.e., the instantaneous rate of change) is defined in terms of a limit and because it is possible that a limit "does not exist" (i.e., as a finite, real number), there are scenarios when [b]a derivative may fail to exist[/b]. This simply means that no finite, numerical value can be assigned as the function's instantaneous rate at that point. [br][br]The term [b]differentiation [/b]refers to [i]the process of finding a derivative[/i]. When such a derivative exists (i.e., as a finite real number), we say that the function is [b]differentiable [/b]at that point, because we are [i]able [/i]to [i]differentiate [/i]it. [br][br]As the examples in this applet demonstration, a function can fail to be differentiable for many reasons and it often boils down to examining the one-sided limits of the difference quotient. If you move on to the next applet (on [i]Linearization[/i]), you will see that differentiability is related to a concept called [b]local linearity[/b], i.e., if you zoom in close enough to a point on the graph of a [i]differentiable [/i]function, the graph will appear to be linear (in fact it will look nearly identical to its tangent line at that point).