Let's look at the Derivative simply for what it is - a function that gives us the slope of another function.

The basic idea of the derivative is actually pretty simple - it's the function [color=#0000ff][math]f'[/math][/color] that gives the [i]slope[/i], rather than the [i]height[/i] ([math]y[/math]-value), of a function [color=#009900][math]f[/math][/color] at each value of [math]x[/math]. One way to think about this is that if the point on the graph of [color=#009900][math]f[/math][/color] at [color=#ff0000][math]x[/math][/color] is [math]([/math][color=#ff0000][math]x[/math][/color], [color=#009900][math]y[/math]-value of [math]f[/math] at [/color][color=#ff0000][math]x[/math][/color][math])[/math], then the point on the graph of [color=#0000ff][math]f'[/math][/color] at [color=#ff0000][math]x[/math][/color] would be [math]([/math][color=#ff0000][math]x[/math][/color], [color=#0000ff]slope of [/color][color=#009900][math]f[/math][/color][color=#0000ff] at [/color][color=#ff0000][math]x[/math][/color][math])[/math].[br][br]As you work with this app, check out Khan Academy's Derivative Intuition Module at [url]https://www.khanacademy.org/math/differential-calculus/taking-derivatives/visualizing-derivatives-tutorial/v/derivative-intuition-module[/url]. This app is based on the same concept, but it allows [b]YOU[/b] to move the dots, and also lets you try many different functions.[br][br]To start with, be sure the "[color=#000000]Show Derivative[/color]" checkbox is cleared ([i]un[/i]checked). The graph of [color=#009900][math]f[/math][/color] is shown in green. Different functions can be investigated by selecting one from the drop-down list at the top right. You can even enter your own "user" function in the "[color=#000000]f(x) =[/color]" box; it will then appear at the bottom of the drop-down list. Now the task is to slide the [color=#ff0000]red dots[/color] up or down so that the [color=#ff0000]red line segment[/color] at that value of [math]x[/math] is [i]tangent to[/i] (matches the slope of) [color=#009900][math]f[/math][/color] at that point. The [math]y[/math]-value of the [color=#ff0000]red dot[/color] is the slope of the corresponding [color=#ff0000]red tangent line segment[/color]. Since the derivative function [color=#0000ff][math]f'[/math][/color] gives us the slope of the original function [color=#009900][math]f[/math][/color], the red dot should therefore lie on the graph of [color=#0000ff][math]f'[/math][/color]. Adjust all the dots so that all the segments are tangent to the green [color=#009900][math]f[/math][/color] graph.[br][br]Now check the "[color=#000000]Show Derivative[/color]" box. The graph of [color=#0000ff][math]f'[/math][/color], the derivative of [color=#009900][math]f[/math][/color], will appear in blue. If you were successful in adjusting the tangent line slopes, your [color=#ff0000]red dots[/color] should lie on or very close to the graph of [color=#0000ff][math]f'[/math][/color].[br][br]A special function is [math]f(x)=e^x[/math]. This is the second-to-last function in the drop-down box. You can drag the graph down and/or use the Zoom buttons to see more points. Make note of the result you get for this function - you'll see this many more times in this course!