## The Chain Rule

The Chain Rule tells us how to differentiate composite functions and since so many functions can be written as composites, it is a vitally important tool for computing derivatives. Before we write down the Chain Rule, let's think about our earlier example.

An Example
 Previously, we considered the composite of two linear functions: and . We found that the composite was . Since these are linear functions, their derivatives are constants--that is, they do not vary from point to point. Notice that The relationship between these numbers is no coincidence. To see why, let's consider the rate of change of the variable in terms of . We will choose two points and . These points also give us Notice that we have Since the graphs of these function are just straight lines, we have related the slopes and hence the derivatives.

Another Example
We'll begin by recalling the following fact from high school. Suppose that is some positive number. Then the graph of the function is obtained from the graph of by compressing the axis by a factor of . The following demonstration will illustrate this fact. If you move the red ball in the rightmost graph, you can see how the function is changed. Also shown are the tangent lines at two points.