The graph on the left shows a function with input x and output u = g(x). That output u is then treated as an input for the function on the right to produce an output y = f(u). But since u = g(x), we could say y = f(g(x)). [br][br][list][*]Use the slider tool for [math]\Delta x[/math] to observe a change in the input x. [/*][*]Use the slider tool for [math]\Delta u[/math] to observe a change in u. Notice that u is an output for g and an input for f. So, you should see a change in both graphs. [/*][*]Use the slider tool for [math]\Delta y[/math] to observe a change in the output y. [/*][/list]Average rates of change will be displayed for each relevant combination of inputs and outputs.

The [b]Chain Rule[/b] (for differentiation) is a rule that explains how the [i]differentiation operator[/i] [math]\frac{d}{dx}[/math] interacts with [b]function composition[/b]: [br][br][math]\frac{d}{dx}\left[f(g(x))\right]=f'(g(x))\cdot g'(x)[/math][br][br]The function notation can be a little cumbersome and difficult to read. If we introduce a substitution variable and think of the composition as a chain of functions, i.e., [math]x\to u\to y[/math], we can simplify the notation a little bit. Let u = g(x) so that y = f(g(x)) = f(u). Then, using Leibniz notation,[br][br][math]\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}[/math]