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 "[math]h\to0[/math]" and observe the relationship between the secant and tangent lines. [/*][/list]

As we have seen, the instantaneous rate of change of a quantity can be estimated using the average rate of change over a small interval. And we define the instantaneous rate of change to be the limit of those average rates of change as the width of the interval shrinks to 0. [br][br]The [b]derivative [/b]of a function [math]f[/math] at a point [math]c[/math] in its domain is the [b]instantaneous rate of change[/b] of [math]f[/math] at [math]x=c[/math]. Because we previously introduced instantaneous rates of change as limits of average rates of change, we have a formula for calculating the derivative at [math]c[/math], denoted by [math]f'(c)[/math]. [br][br]Version 1: [math]f'(c)=\lim_{x\to c}\frac{f(x)-f(c)}{x-c}[/math][br][br]Version 2: [math]f'(c)=\lim_{h\to0}\frac{f(c+h)-f(c)}{h}[/math]