C.2 Chain Rule - Basic Practice
The goal of this page is for you to practice using the Chain Rule on some beginning problems (all free, of course!), each with a complete solution. We'll move to more complex problems on the next page, but you'll probably find that you make some mistakes here, as have we all when first learning this stuff. If so, great! There's no penalty, and most importantly, the only way to learn is through practice, so go ahead and do your early learning right here.
And even though we're calling these "beginning problems," you'll find some toward the bottom that are from past exams at some of the world's best-known science and engineering universities.
We'll start by seeing how the Chain Rule works with the Power Rule, Exponentials, Trig Functions, and then the Product and Quotient Rules.
Chain Rule and Power Rule
Chain Rule and Power Rule
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
Given
View/Hide Solution
Solution 1 (quick, the way most people reason).
Think something like: "The function is
We could of course cancel the 2's, but we're leaving the result as-is so you can easily see how we applied the Chain rule.
Solution 2 (more formal).
Let's use the first form of the Chain rule above:
We have the outer function
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 and Exponentials
Chain Rule and Exponentials
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
Chain Rule and Trig Functions
Chain Rule and Trig Functions
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 and Product 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:
Other Routine Chain Rule Problems
We'll end this screen with some other typical Chain Rule problems you're likely to encounter on an exam โ including a few from actual university exams, which we hope will seem routine after the work you've done above. If they don't seem routine yet, they will soon, as long as you keep practicing!
If Using the Chain Rule at
We'll break this into parts: (1) We'll first find
Subproblem 1: Find
We first need to determine
This value equals the slope of the line segment for
Subproblem 2: Find
Similarly,
Then substituting these values
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:
[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
We'll solve this two ways, first with the Quotient Rule, and then with the Product Rule.
Method 1: Quotient Rule
Method 2: Product Rule
This next problem is a little different, since you're given
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
On the next screen we'll introduce problems that require using the Chain Rule more than once. It's a small step from what we've done above. Since for the rest of the course you're going to need to take such derivatives quickly and correctly, please proceed there to practice as soon as you can.
Do you have questions about any of the problems on this screen, or other Chain Rule problems you're working on? If you post on the Forum, we'll do our best to assist!
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