3-C Chain Rule
Instructions
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)).
- Use the slider tool for to observe a change in the input x.
- Use the slider tool for 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 to observe a change in the output y.
3-C The Chain Rule
The Chain Rule (for differentiation) is a rule that explains how the differentiation operator interacts with function composition:
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., , we can simplify the notation a little bit. Let u = g(x) so that y = f(g(x)) = f(u). Then, using Leibniz notation,