Chain Rule: Problems and Solutions
This screen contains only a summary of how to use the Chain Rule to calculate derivatives, and free problems for you to practice, each with a complete solution immediately available. If you need to get this down quickly, and don't have time to work through the more comprehensive material earlier in this Chapter, this screen should help. These are also great problems to practice for an exam.
Chain Rule Summary
You can always access our Handy Table of Derivatives and Differentiation Rules via the Key Formulas menu item at the top of every page.
You must use the Chain rule to find the derivative of any function that is comprised of one function inside of another function.
For instance,
As another example,
As yet another example,
Since each of these functions is comprised of one function inside of another function — known as a composite function — we must use the Chain rule to find its derivative, as shown in the problems below.
How can I tell what the inner and outer functions are?
Here's a foolproof method: Imagine calculating the value of the function for a particular value of
For example, imagine computing
This imaginary computational process works every time to identify correctly what the inner and outer functions are.
Chain Rule
Consider a composite function whose outer function is
Alternatively, if we write
Informally:
Even though few people admit it, almost everyone thinks along the lines of the informal approach in the blue boxes above. We'll illustrate in the problems below.
Chain Rule Example #1
Differentiate
Solutions.
We'll solve this using three different approaches — but we encourage you to become comfortable with the third approach as quickly as possible, because that's the one you'll use to compute derivatives quickly as the course progresses.
• Solution 1.
Let's use the first form of the Chain rule above:
We have the outer function
Then
We could of course simplify the result algebraically to
• Solution 2.
Let's use the second form of the Chain rule above:
We have
Then
• Solution 3.
Let's use the third form of the Chain rule above:
Instead, you'll think something like: "The function is a bunch of stuff to the 7th power. So the derivative is 7 times that same stuff to the 6th power, times the derivative of that stuff."
Note: You'd never actually write out "stuff = ...." Instead just hold in your head what that "stuff" is, and proceed to write down the required derivatives.
There are lots more completely solved example problems below!
The problems below are grouped for practice purposes to use the Chain rule and: [Power rule] [Exponentials] [Trig Functions] [Product rule & Quotient rule] [Chain rule multiple times] [More problems and University exam problems]
Chain Rule & Power Rule
You'll usually see this written as
The following six problems illustrate.
Solution 1 (quick, the way most people reason).
Think something like: "The function is some stuff to the eighth-power. So the derivative is eight times that same stuff to the seventh power, times the derivative of that stuff."
Note: You'd never actually write out "stuff = ...." Instead just hold in your head what that "stuff" is, and proceed to write down the required derivatives.
Solution 2 (more formal).
Let's use the first form of the Chain rule above:
Then
Given
Hint: Recall
We'll again solve this two ways. The first is the way most experienced people quickly develop the answer, and that we hope you'll soon be comfortable with. The second is more formal.
Solution 1 (quick, the way most people reason).
Think something like: "The function is some stuff to the power of 3. So the derivative is 3 times that same stuff to the power of 2, times the derivative of that stuff."
Note: You'd never actually write out "stuff = ...." Instead just hold in your head what that "stuff" is, and proceed to write down the required derivatives.
Solution 2 (more formal).
Let's use the first form of the Chain rule above:
Then
Hence
Solution 1 (quick, the way most people reason).
Think something like: "The function is some stuff to the
Note: You'd never actually write out "stuff = ...." Instead just hold in your head what that "stuff" is, and proceed to write down the required derivatives.
Solution 2 (more formal).
Let's use the first form of the Chain rule above:
We have the outer function
Then
Hence
Think something like: "The function is some stuff to the 99th power. So the derivative is 99 times that same stuff to the 98th power, times the derivative of that stuff."
Solution 2 (more formal).
Let's use the first form of the Chain rule above:
Then
Hence
Think something like: "The function is
Solution 2.
Let's use the first form of the Chain rule above:
Then
Solution 1 (quick, the way most people reason).
Think something like: "The function is
Solution 2 (more formal).
Let's use the first form of the Chain rule above:
Then
Chain Rule & Exponentials
You'll usually see this written as
The next three problems illustrate.
Given
Solution 1 (quick, the way most people reason).
Think something like: "The function is
Note: You'd never actually write out "stuff = ...." Instead just hold in your head what that "stuff" is, and proceed to write down the required derivatives.
Solution 2 (more formal).
Let's use the first form of the Chain rule above:
Then
Solution 1 (quick, the way most people reason).
Think something like: "The function is
Solution 2 (more formal).
Let's use the first form of the Chain rule above:
Then
Hence
Think something like: "The function is
Solution 2 (more formal).
Let's use the first form of the Chain rule above:
Then
Chain Rule & Trig Functions
You'll usually see this written as
You'll usually see this written as
You'll usually see this written as
The next two problems illustrate.
Solution 1.
Think something like: "The function is sin(of some stuff). So the derivative is cos(of that same stuff), times the derivative of that stuff."
Solution 2.
Let's use the first form of the Chain rule above:
Then
Given
Solution 1.
Think something like: "The function is tan(of some stuff). So the derivative is
Solution 2.
Let's use the first form of the Chain rule above:
We have the outer function
Then
Chain Rule & Product Rule or Quotient Rule
The next few problems require using the Chain rule with the Product rule or with the Quotient rule.
This problem combines the Product Rule with the Chain Rule.
Given
(A)
(B)
(C)
(D)
(E) none of these
Let's first think about the derivative of each term separately. We won't write out "stuff" as we did before to use the Chain Rule, and instead will just write down the answer using the same thinking as above:
and
We instead stopped where we did above to emphasize the way we've developed the result, which is what matters most here. (You don't need us to show you how to do algebra!)
Besides, on an exam your grader is most likely to check for something that looks like our result, which shows that you know how to use both the Product and Chain Rules correctly. You might ask your teacher how much you should simplify on an exam: small algebraic mistakes often happen during the "simplification process," so if you won't lose points by not simplifying at all, that's your best bet.
This problem combines the Quotient rule with the Chain rule.
(A)
(B)
(C)
(D)
(E) none of these
First, the numerator:
Using the Chain Rule Multiple Times
The next few problems require using the Chain rule multiple times.
This problem requires using the Chain Rule twice.
Differentiate
Solution 1 (quick, the way most people reason).
Think something like: The overall function is
The outermost function is thus
Hence we first write
Solution 2 (more formal).
Although it's tedious to write out each separate function, let's use an extension of the first form of the Chain rule above, now applied to
Then
This problem requires using the Chain Rule twice.
Differentiate
Solution 1 (quick, the way most people reason).
Think something like:
The overall function is
The outermost function is thus
Next, the first inside function is
Solution 2 (more formal).
Although it's tedious to write out each separate function, let's use an extension of the first form of the Chain rule above, now applied to
Then
This problem requires using the Chain Rule three times.
DifferentiateHint: Recall that
We won't write out all of the tedious substitutions, and instead reason the way you'll need to become comfortable with:
The outermost function is
More Problems and University Exam Problems
The following problems use the Chain Rule in different ways, and contain some actual university exam problems.
Let
Find
The key realization in this problem is that the requested limit is the definition of the derivative of
Once we realize that, we can use what we know about the Chain rule and read the necessary values from the table:
Show that
Hint: Start with
Start from the hint, and recall that
The solution is easiest to understand if we rewrite the given derivative as
Because now we can invoke the Chain Rule, using
A less formal approach, using "stuff":
We can view the function
[This problem appeared on an exam at a well-known science and engineering university.]
Differentiate
[This problem appeared on an exam at a well-known science and engineering university.]
Differentiate
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