[b]Right: [/b]Investigate the secant and tangent lines at a point P on the graph of a function. [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. [/*][*]Use the input box for h or the button "[math]h\to0[/math]" to move the point Q around P. [/*][*]The "Secant" and "Tangent" checkboxes will show/hide the respective lines. [/*][*]The "Difference Quotient" checkbox will show/hide the graph of the difference quotient function, which has an excluded value when h = 0. [/*][/list][b]Left: [/b]A zoomed in version of the point on the graph and the difference between the change [math]\Delta y[/math] and the differential [math]dy[/math]. [br][list][*]Use the "dx" checkbox to show/hide the horizontal change from x=c to x=c+h. [/*][*]Check the "Secant" box to also reveal a checkbox for [math]\Delta y[/math]. This will show/hide the vertical change along the graph of f from x = c to x=c+h. [/*][*]Check the "Tangent" box to reveal a checkbox for [math]dy[/math]. This will show/hide the vertical change along the tangent line from x=c to x=c+h. [/*][*]Check the "Error" box to highlight the difference between [math]\Delta y[/math] and [math]dy[/math]. [/*][/list]

As mentioned in the previous applet (on [i]Differentiability[/i]), the existence of a derivative at a point is closely related to a concept called [b]local linearity[/b]. When a function is differentiable it also has local linearity, and vice versa. If you zoom in close enough to a point on the graph of a differentiable function, the graph will look like a straight line. In fact, it will nearly be identical to its tangent line if you zoom in far enough. [br][br]Implications: Differentiable functions can be approximated with linear functions, called the [b]linearization [/b](i.e., tangent line), as long as you keep your inputs relatively close together. The point where the linearization is generated is called the [b]center [/b]of the linearization. As long as the input is "close enough" to the center, the linearization should give good estimates of nearby function values.