Composition of Functions is a way to chain together so that the output of the first inner function becomes the input of the next outer function. [math]f\circ g\left(x\right)=f\left(g\left(x\right)\right)[/math]. Note that we read this from right to left starting with the input of [i]x[/i] then applying the function [i]g[/i] and finally applying the function [i]f[/i]. The first function produces ([i]x[/i], [i]g[/i]([i]x[/i])). The output becomes the input for the next stage producing ([i]g[/i]([i]x[/i]), [i]f[/i]([i]g[/i]([i]x[/i])). Ultimately the composition function, [i]c[/i]([i]x[/i]) above, is made up of the ordered pairs ([i]x[/i], [i]f[/i]([i]g[/i]([i]x[/i])). [br][br]In the app enter a formula for the first inner function [i]g[/i]([i]x[/i]) in terms of [i]x[/i]. It is graphed in the [i]xy[/i]-plane in blue in the left window. Next enter a formula for the second outer function [i]f[/i]([i]u[/i]) in terms of [i]u[/i]. It is graphed in the [i]uy[/i]-plane in green in the right window. Adjust the value of the specific value of [i]x[/i] = [i]a[/i]. This is indicated by the brown length. The intermediate output [i]g[/i]([i]a[/i]) is indicated by the purple length. This is an output [i]y[/i]-value (vertical length) in the left window, but it becomes an input [i]u[/i]-value (horizontal length) in the right window. The output [i]f[/i]([i]g[/i]([i]a[/i])) is illustrated in orange as the output in both windows for [i]f[/i] and [i]c[/i].

The Chain Rule (official name) could also be called the Composition Rule, since it is for finding the derivative of a composition of two functions. It could also be called the Substitution Rule, because to use it we make a substitution[i] u[/i] = [i]g[/i]([i]x[/i]) to form the function [i]f[/i]([i]u[/i]). [br][br]In the app check the checkbox for Tangent Lines to see the tangent lines to the three function at [i]x[/i] = [i]a[/i] for [i]g[/i] and [i]c [/i]and at [i]u[/i] = [i]g[/i]([i]a[/i]) for [i]f[/i]. The slopes of these three tangent lines are the derivatives, and they are illustrated by the vertical vectors in the app. Values of the three derivatives are shown. Do you see a relationship among these three slopes?[br][br]You might notice that the one we are looking for (the slope of the red tangent line) is the product of the other two.

Check the checkbox for Derivative to see a statement of the Derivative Chain Rule along with formulas and values based on the specific functions and values in the app.[br][br]Check the Proof checkbox to see the steps of a formal proof of the Chain Rule.