This is an illustration of the Chain Rule of Differentiation using two straight lines. The chain rule allows you to take the derivative of composite functions.  In this case two lines are shown representing the linear functions [math]y=mx+b[/math].  The composite function can be easily calculated as [math]y=f\left(g\left(x\right)\right)=m_f\left(m_gx+b_g\right)+b_f=m_fm_bx+m_fb_g+b_f[/math].  The slope of the composite function can be seen to be the product of the slopes of the component functions.  This follows the chain rule [br][math]\frac{df(g(x))}{dx}=\frac{df(g)}{dg}\frac{dg}{dx}[/math][br][br][br][b][size=150]Using the Applet[br][/size][/b][br]The applet also shows the path of the values from the abscissa (independent variable) to the ordinate( dependent variable) through both functions.  The [math]x[/math] value can be changed by dragging the diamond point.[br][br]The equations for the lines can be edited to change the slope and [math]y[/math]-intercepts in the text entries.[br][br]When "Show [math]\Delta[/math]" is checked a slider for [i]h [/i]can be used to adjust a change in the [math]\Delta x[/math] value which is passed through both functions.  A shaded region is used to show how the rise varies when the run is changed.  For example if [math]m=2[/math] the rise is twice as wide as the run.[br][br]When "Show f(g(x))" is checked the single equivalent composite function is shown.  This is the calculated function for [math]y[/math] shown above.  The data path of the data is also shown with dotted lines.

[b][size=150]Things to Do[/size][/b][br]Move the x diamond to see how the data flows through both functions.[br]Change the slope and [math]y[/math]-intercept of both functions in the text boxes.[br]Note how the resulting derivative [math]\frac{df}{dx}[/math] changes with the equation changes.[br][br]Show [math]\Delta[/math] and note how wide the shapes are on the abscissa and ordinate as the [math]\Delta[/math] passes through the functions.  Caution you can change the axis values to distort the widths.[br][br]Show f(g(x)) to see the composite function.  Note the ordinate values and the shape widths.  The "Show [math]\Delta[/math]" can be turned on or off to clean up the view.