AP Calculus (AB)
The Chain Rule

Many students find the Chain Rule somewhat difficult. At an AP Calculus Institute in July 2002 the presenter demonstrated a method of using the Chain Rule that seemed to streamline the process and may aid you in solving derivative problems that involve a function composition.

First, the "Chain Rule" is one of the most widely used differentiation techniques. It is necessary in derivatives that involve a composition of functions that cannot be expanded out to something simpler. For example, the following are all function compositions but do not necessarily need the Chain Rule because the "composition" can be expanded out to something for which we already have differentiation rules or the derivative can be found from the quotient rule or product rule.

However, the following can only be expanded out by recalling a trigonometric identity. It turns out to be much easier to differentiate with the Chain Rule.

How do we know the derivative is not simply ? Well, we could try graphing the function along with its numerical derivative and observe that the proposed derivative is not correct.

     

The derivative appears similar to but it has an amplitude of 2, not 1. Obviously, something else is going on to obtain the derivative.

The definition of the Chain Rule is, given with , then .

Thus, given , note the composition. The "trick" to using the chain rule with more accuracy is to avoid the stress of defining u and the "textbook" order of the derivative and to follow the simple pattern:

derivative of what's happening to the inner-most x ... times ... the derivative of the "rest"

This is terrible formal math ... but it should get you over the hurdle of the Chain Rule ... and with practice fuller understanding should follow. Thus, the derivative of is

This method works particularly well with problems involving a "double-chain rule." Consider:

This is really the composition of three functions. From the inside out we have:

then and then finally

Therefore, to get the derivative we take the derivative of the inner most piece, multiply it by the derivative of the next piece, and then multiply that by the next (here, the last) piece.

Often we will be able to simplify the results from applying the Chain Rule. This is necessary sometimes if we need to set the derivative equal to zero to find critical numbers. As a student, you will need to match answers with multiple choice selections, thus you may need to simplify to match your answer with the correct option. Here, g' simplifies to:

Each of these are equivalent. I prefer the final form because it would be simplest (for me) to set equal to zero and find where the derivative equals zero or is undefined.

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Additional Links

• Differentiation with the Chain Rule
• MathWorld - Definition of the Chain Rule