Chain Rule Help

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:

(1)

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.