The chain rule is similar to the product rule and the quotient rule, but it deals with differentiating compositions of functions. If we recall, a composite function is a function that contains another function:
The Formula for the Chain Rule
The capital F means the same thing as lower case f, it just encompasses the composition of functions. As a motivation for the chain rule, let's look at the following example:
This function would take a long time to factor out and find the derivative of each term, so we can consider this a composite function. The two functions would look like this:
Notice that substituting g(x) for g in f(x) would yeild the original function. We will see that after differentiating, we will then substitute g(x) back in for g.
So the composite function would be
Now, we can use the chain rule, which is defined by taking the derivative of outside function times the inside function, and multiplying it by the derivative of the inside function:
Using this rule, we have:
Let's do another example.
(2) Differentiate the following function:
We define the inside and outside function to be
Then, the derivative of the composition will be as follows:
Think of the chain rule as a process. The derivative of the composite function is the derivative of the outside function times the derivative of the inside function.