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

If๐‘“(๐‘ฅ)=(stuff)๐‘›,then๐‘‘๐‘“๐‘‘๐‘ฅ=๐‘›(that stuff)๐‘›โˆ’1โ‹…๐‘‘๐‘‘๐‘ฅ(that stuff) You'll usually see this written as ๐‘‘๐‘‘๐‘ฅ(๐‘ข๐‘›)=๐‘›๐‘ข๐‘›โˆ’1โ‹…๐‘‘๐‘ข๐‘‘๐‘ฅ The following problems illustrate and let you practice.

Chain Rule and Power Rule Problem #1
Given ๐‘“(๐‘ฅ) =(3๐‘ฅ2โˆ’4๐‘ฅ+5)8, ๐‘“โ€ฒ(๐‘ฅ) = (A) 8(3๐‘ฅ2โˆ’4๐‘ฅ+5)7(B) 8(3๐‘ฅ2โˆ’4๐‘ฅ+5)7โ‹…(6๐‘ฅโˆ’4)(C) 8(6๐‘ฅโˆ’4)7 (D) (3๐‘ฅ2โˆ’4๐‘ฅ+5)8(E) none of these
Chain Rule and Power Rule Problem #2
Given ๐‘“(๐‘ฅ) =tan3โก๐‘ฅ, ๐‘“โ€ฒ(๐‘ฅ) =Hint: Recall tan3โก๐‘ฅ =[tanโก๐‘ฅ]3. Also recall that ๐‘‘๐‘‘๐‘ฅtanโก๐‘ฅ =sec2โก๐‘ฅ.(A) 3sec4โก๐‘ฅ(B) 3tan2โก๐‘ฅ(C) tan3โก๐‘ฅsec2โก๐‘ฅ (D) 3tan2โก๐‘ฅโ‹…sec2โก๐‘ฅ(E) none of these
Chain Rule and Power Rule Problem #3

Given ๐‘“(๐‘ฅ) =โˆš๐‘ฅ2+1, ๐‘“โ€ฒ(๐‘ฅ) =

(A) 121โˆš๐‘ฅ2+1โ‹…(2๐‘ฅ)(B) 121โˆš๐‘ฅ2+1(C) 121โˆš2๐‘ฅ (D) 12โˆš๐‘ฅ2+1(E) none of these

View/Hide Solution

Solution 1 (quick, the way most people reason).

Think something like: "The function is โˆšstuff. So the derivative is 121โˆšthat same stuff, times the derivative of that stuff."

๐‘‘๐‘“๐‘‘๐‘ฅ=[๐‘‘๐‘“๐‘‘(stuff), with the same stuff inside]ร—๐‘‘๐‘‘๐‘ฅ(stuff) ๐‘“(๐‘ฅ)=โˆšstuff;stuff=๐‘ฅ2+1Then๐‘“(๐‘ฅ)=๐‘‘๐‘“๐‘‘๐‘ฅ=121โˆšstuffโ‹…(๐‘‘๐‘‘๐‘ฅ(stuff))=121โˆš๐‘ฅ2+1โ‹…(2๐‘ฅ)โŸน (A)โœ“

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:

[๐‘“(๐‘”(๐‘ฅ))]โ€ฒ=๐‘“โ€ฒ(๐‘”(๐‘ฅ))โ‹…๐‘”โ€ฒ(๐‘ฅ)=[derivative of the outer function, evaluated at the inner function] ร— [derivative of the inner function]

We have the outer function ๐‘“(๐‘ข) =โˆš๐‘ข and the inner function ๐‘ข =๐‘”(๐‘ฅ) =๐‘ฅ2 +1.

Then (โˆš๐‘ข)โ€ฒ =121โˆš๐‘ข, and (๐‘ฅ2+1)โ€ฒ =2๐‘ฅ.

[close solution]

Chain Rule and Power Rule Problem #4
Given ๐‘“(๐‘ฅ) =โˆšsinโก๐‘ฅ, ๐‘“โ€ฒ(๐‘ฅ) = (A) 121โˆšsinโก๐‘ฅ(B) 12โˆšsinโก๐‘ฅโ‹…cosโก๐‘ฅ(C) 121โˆšsinโก๐‘ฅโ‹…cosโก๐‘ฅ(D) โˆ’121โˆšsinโก๐‘ฅโ‹…cosโก๐‘ฅ(E) none of these

Chain Rule and Exponentials

Chain Rule and Exponentials

If๐‘“(๐‘ฅ)=๐‘’(stuff),then๐‘‘๐‘“๐‘‘๐‘ฅ=๐‘’(that stuff)โ‹…๐‘‘๐‘‘๐‘ฅ(that stuff) You'll usually see this written as ๐‘‘๐‘‘๐‘ฅ๐‘’๐‘ข=๐‘’๐‘ขโ‹…๐‘‘๐‘ข๐‘‘๐‘ฅ The following problems illustrate and let you practice.

Chain Rule and Exponentials Problem #1
Given ๐‘“(๐‘ฅ) =๐‘’sinโก๐‘ฅ, ๐‘“โ€ฒ(๐‘ฅ) =(A) ๐‘’cosโก๐‘ฅcosโก๐‘ฅ(B) ๐‘’(sinโก๐‘ฅโˆ’1)๐‘’cosโก๐‘ฅ(C) ๐‘’โˆ’sinโก๐‘ฅcosโก๐‘ฅ (D) ๐‘’sinโก๐‘ฅcosโก๐‘ฅ(E) none of these

Chain Rule and Exponentials Problem #2
Given ๐‘“(๐‘ฅ) =๐‘’๐‘ฅ2, ๐‘“โ€ฒ(๐‘ฅ) = (A) ๐‘’๐‘ฅ2๐‘’2๐‘ฅ(B) ๐‘’๐‘ฅ2โ‹…2๐‘ฅ(C) ๐‘’๐‘ฅ2(D) ๐‘’2๐‘ฅ(E) none of these

Chain Rule and Trig Functions

Chain Rule and Trig Functions

If๐‘“(๐‘ฅ)=sinโก(stuff),then๐‘‘๐‘“๐‘‘๐‘ฅ=cosโก(that stuff)โ‹…๐‘‘๐‘‘๐‘ฅ(that stuff) You'll usually see this written as ๐‘‘๐‘‘๐‘ฅsinโก๐‘ข=cosโก๐‘ขโ‹…๐‘‘๐‘ข๐‘‘๐‘ฅ

If๐‘“(๐‘ฅ)=cosโก(stuff),then๐‘‘๐‘“๐‘‘๐‘ฅ=โˆ’sinโก(that stuff)โ‹…๐‘‘๐‘‘๐‘ฅ(that stuff) You'll usually see this written as ๐‘‘๐‘‘๐‘ฅcosโก๐‘ข=โˆ’sinโก๐‘ขโ‹…๐‘‘๐‘ข๐‘‘๐‘ฅ
If๐‘“(๐‘ฅ)=tanโก(stuff),then๐‘‘๐‘“๐‘‘๐‘ฅ=sec2โก(that stuff)โ‹…๐‘‘๐‘‘๐‘ฅ(that stuff) You'll usually see this written as ๐‘‘๐‘‘๐‘ฅtanโก๐‘ข=sec2โก๐‘ขโ‹…๐‘‘๐‘ข๐‘‘๐‘ฅ The following problems illustrate and let you practice.

Chain Rule and Trig Functions Problem #1
Given ๐‘“(๐‘ฅ) =sinโก(2๐‘ฅ), ๐‘“โ€ฒ(๐‘ฅ) = (A) cosโก(2๐‘ฅ)โ‹…(2)(B) sinโก(2๐‘ฅ)โ‹…(2)(C) โˆ’cosโก(2๐‘ฅ)โ‹…(2) (D) cosโก(2)(E) none of these
Chain Rule and Trig Functions Problem #2
Given ๐‘“(๐‘ฅ) =tanโก(๐‘’๐‘ฅ), ๐‘“โ€ฒ(๐‘ฅ) = (A) secโก(๐‘’๐‘ฅ)tanโก(๐‘’๐‘ฅ)โ‹…๐‘’๐‘ฅ(B) secโก(๐‘’๐‘ฅ)โ‹…๐‘’๐‘ฅ(C) sec2โก(๐‘’๐‘ฅ) (D) sec2โก(๐‘’๐‘ฅ)โ‹…๐‘’๐‘ฅ(E) none of these

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.

Chain & Product, Quotient Rule Problem #1
This problem combines the Product Rule with the Chain Rule. Given ๐‘“(๐‘ฅ) =(๐‘ฅ2+1)7(3๐‘ฅ โˆ’7)4, ๐‘“โ€ฒ(๐‘ฅ) = (A) 28(๐‘ฅ2+1)6(3๐‘ฅ โˆ’7)3(B) 28(๐‘ฅ2+1)6(3๐‘ฅ โˆ’7)3 โ‹…(2๐‘ฅ) โ‹…(3)(C) [7(๐‘ฅ2+1)6โ‹…(2๐‘ฅ)](3๐‘ฅ โˆ’7)4 +(๐‘ฅ2+1)7[4(3๐‘ฅโˆ’7)3โ‹…(3)](D) 7(๐‘ฅ2+1)6(3๐‘ฅ โˆ’7)4 +4(๐‘ฅ2+1)7(3๐‘ฅ โˆ’7)3(E) none of these
Chain & Product, Quotient Rule Problem #2
This problem combines the Quotient rule with the Chain rule.Given โ„Ž(๐‘ฅ) =๐‘’2๐‘ฅ1โˆ’๐‘ฅ2, ๐‘“โ€ฒ(๐‘ฅ) =(A) (๐‘’2๐‘ฅโ‹…2)(1โˆ’๐‘ฅ2)โˆ’(๐‘’2๐‘ฅ)(โˆ’2๐‘ฅ)(1โˆ’๐‘ฅ2)2(โˆ’2๐‘ฅ) (B) (๐‘’2๐‘ฅโ‹…2)(1โˆ’๐‘ฅ2)โˆ’(๐‘’2๐‘ฅ)(โˆ’2๐‘ฅ)(1โˆ’๐‘ฅ2)2(C) (๐‘’2๐‘ฅโ‹…2)(1โˆ’๐‘ฅ2)โˆ’2(๐‘’2๐‘ฅ)(1โˆ’๐‘ฅ2)2 (D) (๐‘’2๐‘ฅโ‹…2)(1โˆ’๐‘ฅ2)+(๐‘’2๐‘ฅ)(โˆ’2๐‘ฅ)(1โˆ’๐‘ฅ2)2 (E) none of these

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!

Chain Rule General Problem #1
Given that ๐‘“(2) =1, ๐‘“โ€ฒ(4) =5, ๐‘”(2) =4, and ๐‘”โ€ฒ(2) =8, find [๐‘“(๐‘”(2))]โ€ฒ.(A) 5(B) 10(C) 40(D) 8(E) 32
Chain Rule General Problem #2
Graphs of piecewise functions f and g, each of which consists of line segments. The graph of f has junction points at (-3.5, 3.5), (-1.5, -0.5), (1.5, 2.5), and (3, -2). The graph of g has junction points at (-3, -2), (-1, 2), (1, 2), and (5, -2). If ๐‘“(๐‘ฅ) and ๐‘”(๐‘ฅ) are represented in the graph, find [๐‘“(๐‘”(๐‘ฅ))]โ€ฒ at ๐‘ฅ =0. (A) 0(B) โˆ’1(C) 1(D) โˆ’32(E) 12
Chain Rule General Problem #3
Let ๐‘“ and ๐‘” be differentiable functions and let the values of ๐‘“,๐‘”,๐‘“โ€ฒ and ๐‘”โ€ฒ at ๐‘ฅ =1 and ๐‘ฅ =2 be given by the table. ๐‘ฅ๐‘“(๐‘ฅ)๐‘”(๐‘ฅ)๐‘“โ€ฒ(๐‘ฅ)๐‘”โ€ฒ(๐‘ฅ)153272โˆ’2146 Find limโ„Žโ†’0๐‘“(๐‘”(2+โ„Ž))โˆ’๐‘“(๐‘”(2))โ„Ž.
Chain Rule General Problem #4
[This problem appeared on an exam at a well-known science and engineering university.]Differentiate ๐‘“(๐‘ฅ) =sinโก(2๐‘ฅ)๐‘ฅ.
Chain Rule General Problem #5
[This problem appeared on an exam at a well-known science and engineering university.]Differentiate ๐‘”(๐‘ฅ) =๐‘ฅ2โˆš1โˆ’๐‘ฅ.

This next problem is a little different, since you're given ๐‘“โ€ฒ(๐‘ฅ) and then asked to find the derivative of ๐‘“(๐‘ฅ2). You may encounter a similar problem in your homework or on an exam; this problem was taken, in fact, from an exam at a well-known university.

Chain Rule General Problem #6 (prior uni exam question)
If ๐‘ฆ =๐‘“(๐‘ฅ2) and ๐‘“โ€ฒ(๐‘ฅ) =โˆš3๐‘ฅ+5, show that ๐‘‘๐‘ฆ๐‘‘๐‘ฅ =2๐‘ฅโˆš3๐‘ฅ2+5.

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!