We have covered almost all of the derivative rules that deal with combinations of two (or more) functions. The operations of addition, subtraction, multiplication (including by a constant) and division led to the Sum and Difference rules, the Constant Multiple Rule, the Power Rule, the Product Rule and the Quotient Rule. To complete the list of differentiation rules, we look at the last way two (or more) functions can be combined: the process of composition (i.e. one function “inside” another).
One example of a composition of functions is . We currently do not know how to compute this derivative. If forced to guess, one would likely guess , where we recognize as the derivative of and as the derivative of . However, this is not the case; . In Example 2.5.4 we’ll see the correct answer, which employs the new rule this section introduces, the Chain Rule.
Before we define this new rule, recall the notation for composition of functions. We write or , read as “ of of ,” to denote composing with . In shorthand, we simply write or and read it as “ of .” Before giving the corresponding differentiation rule, we note that the rule extends to multiple compositions like or , etc.
To motivate the rule, let’s look at three derivatives we can already compute.
Find the derivatives of , and (We’ll see later why we are using subscripts for different functions and an uppercase .)
SolutionIn order to use the rules we already have, we must first expand each function as , and .
It is not hard to see that:
An interesting fact is that these can be rewritten as
A pattern might jump out at you. Recognize that each of these functions is a composition, letting :
We’ll come back to this example after giving the formal statements of the Chain Rule; for now, we are just illustrating a pattern.
Let be a differentiable function of and let be a differentiable function of . Then is a differentiable function of , and
We can think of this as taking the derivative of the outer function evaluated at the inner function times the derivative of the inner function. To help understand the Chain Rule, we return to Example 2.5.1.
Use the Chain Rule to find the derivatives of the functions given in Example 2.5.1.
SolutionExample 2.5.1 ended with the recognition that each of the given functions was actually a composition of functions. To avoid confusion, we ignore most of the subscripts here.
We found that
To find , we apply the Chain Rule. We need and
Part of the Chain Rule uses . This means substitute for in the equation for . That is, . Finishing out the Chain Rule we have
Let , where and . We have , so . The Chain Rule then states
Finally, when , we have and . Thus and . Thus
Example 2.5.2 demonstrated a particular pattern: when and , then . This is called the Generalized Power Rule.
Let be a differentiable function. Then
This allows us to quickly find the derivative of functions like . While it may look intimidating, the Generalized Power Rule states that
Treat the derivative-taking process step-by-step. In the example just given, first multiply by 20, then rewrite the inside of the parentheses, raising it all to the 19 power. Then think about the derivative of the expression inside the parentheses, and multiply by that.
Watch the video:
Chain Rule for Finding Derivatives from https://youtu.be/6kScLENCXLg
We now consider more examples that employ the Chain Rule.
Find the derivatives of the following functions:
Consider . Recognize that this is a composition of functions, where and . Thus
Recognize that is the composition of and . Also, recall that
This leads us to:
Recognize that is the composition of and . Remembering that , we have
Let . Find the equation of the line tangent to the graph of at .
SolutionThe tangent line goes through the point with slope . To find , we need the Chain Rule.
. Evaluated at , we have . Thus the equation of the tangent line is approximately ††margin: Λ
The tangent line is sketched along with in Figure 2.5.1.
The Chain Rule is used often in taking derivatives. Because of this, one can become familiar with the basic process and learn patterns that facilitate finding derivatives quickly. For instance,
A concrete example of this is
While the derivative may look intimidating at first, look for the pattern. The denominator is the same as what was inside the natural log function; the numerator is simply its derivative.
This pattern recognition process can be applied to lots of functions. In general, instead of writing “anything”, we use as a generic function of . We then say
The following is a short list of how the Chain Rule can be quickly applied to familiar functions.
Of course, the Chain Rule can be applied in conjunction with any of the other rules we have already learned. We practice this next.
Find the derivatives of the following functions.
We must use the Product and Chain Rules. Do not think that you must be able to “see” the whole answer immediately; rather, just proceed step-by-step.
We must employ the Quotient Rule along with the Chain Rule. Again, proceed step-by-step.
A key to correctly working these problems is to break the problem down into smaller, more manageable pieces. For instance, when using the Product and Chain Rules together, just consider the first part of the Product Rule at first: . Just rewrite , then find . Then move on to the part. Don’t attempt to figure out both parts at once.
Likewise, using the Quotient Rule, approach the numerator in two steps and handle the denominator after completing that. Only simplify afterward.
We can also employ the Chain Rule itself several times, as shown in the next example.
Find the derivative of .
SolutionRecognize that we have the function “inside” the function ; that is, we have . We use the Chain Rule multiple times, beginning with the Generalized Power Rule:
This function is frankly a ridiculous function, possessing no real practical value. It is very difficult to graph, as the tangent function has many vertical asymptotes and grows so very fast. The important thing to learn from this is that the derivative can be found. In fact, it is not “hard;” one must take several small steps and be careful to keep track of how to apply each of these steps.
It is a traditional mathematical exercise to find the derivatives of arbitrarily complicated functions just to demonstrate that it can be done. Just break everything down into smaller pieces.
Find the derivative of .
SolutionThis function likely has no practical use outside of demonstrating derivative skills. The answer is given below without simplification. It employs the Quotient Rule, the Product Rule, and the Chain Rule three times.
The reader is highly encouraged to look at each term and recognize why it is there. This example demonstrates that derivatives can be computed systematically, no matter how arbitrarily complicated the function is.
It is instructive to understand what the Chain Rule “looks like” using “” notation instead of notation. Suppose that is a function of , where is a function of , as stated in Theorem 2.5.1. Then, through the composition , we can think of as a function of , as . Thus the derivative of with respect to makes sense; we can talk about This leads to an interesting progression of notation:
|(since and )|
|(using “fractional” notation for the derivative)|
Here the “fractional” aspect of the derivative notation stands out. On the right hand side, it seems as though the “” terms divide out, leaving
It is important to realize that we are not dividing these terms; the derivative notation of is one symbol. It is equally important to realize that this notation was chosen precisely because of this behavior. It makes applying the Chain Rule easy with multiple variables. For instance,
where and are any variables you’d like to use.
One of the most common ways of “visualizing” the Chain Rule is to consider a set of gears, as shown in Figure 2.5.2. The gears have 36, 18, and 6 teeth, respectively. That means for every revolution of the gear, the gear revolves twice. That is, the rate at which the gear makes a revolution is twice as fast as the rate at which the gear makes a revolution. Using the terminology of calculus, the rate of -change, with respect to , is .
Likewise, every revolution of causes 3 revolutions of : . How does change with respect to ? For each revolution of , revolves 6 times; that is,
We can then extend the Chain Rule with more variables by adding more gears to the picture.
It is difficult to overstate the importance of the Chain Rule. So often the functions that we deal with are compositions of two or more functions, requiring us to use this rule to compute derivatives. It is often used in practice when actual functions are unknown. Rather, through measurement, we can calculate and . With our knowledge of the Chain Rule, finding is straightforward.
In the next section, we use the Chain Rule to justify another differentiation technique. There are many curves that we can draw in the plane that fail the “vertical line test.” For instance, consider , which describes the unit circle. We may still be interested in finding slopes of tangent lines to the circle at various points. The next section shows how we can find without first “solving for .” While we can in this instance, in many other instances solving for is impossible. In these situations, implicit differentiation is indispensable.
T/F: The Chain Rule describes how to evaluate the derivative of a composition of functions.
T/F: The Generalized Power Rule states that .
T/F: Taking the derivative of requires the use of both the Product and Chain Rules.
In Exercises 7–46., compute the derivative of the given function.
If with , , , and . Find .
Suppose , where , , , , and . Find .
In Exercises 51–54., find the equations of tangent line to the graph of the function at the given point. Note: the functions here are the same as in Exercises 7.–10..
Use the Chain Rule and Product Rule to give an alternative proof of the Quotient Rule. (Hint: write as .
Use the Chain Rule to express the second derivative of in terms of first and second derivatives of and .
Find the derivative of