Worksheets · AP Calculus AB

AP Calculus AB Unit 3 Worksheet: Chain Rule, Implicit, Inverse

20 exam-style multiple-choice questions on the chain rule, implicit differentiation and inverse functions, free to use in class. Good practice before the Unit 3 Progress Check in AP Classroom. Free response for this unit: Unit 3 FRQ set.

Sigma Prep · sigmaprep.io/worksheets/ap-calculus-ab

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AP Calculus AB Unit 3: Chain Rule, Implicit, Inverse

Date Version A

Directions: Work each problem in the space below it, then choose the best of the four answer choices. Unless a question says otherwise, the domain of a function f is every real number x for which f(x) is a real number.

Part A: No calculator is allowed.

Questions 1–19
  1. How fast is f(x)=cos⁡(3x) changing with respect to x at x=5π18?
    1. (A)−12
    2. (B)−332
    3. (C)32
    4. (D)−32
  2. The table gives values of an increasing, differentiable function f and of its derivative f′. If g is the inverse function of f, what is g′(2)?
    x−2−1012f(x)−1261013f′(x)84633
    1. (A)13
    2. (B)−14
    3. (C)14
    4. (D)113
  3. What is the derivative of f(x)=e−x−7?
    1. (A)(−x−7)e−x−8
    2. (B)−e−1
    3. (C)(−x−7)e−x−7
    4. (D)−e−x−7
  4. Differentiate 2arctan⁡(2x) with respect to x. The result is
    1. (A)41+4x2
    2. (B)41+2x2
    3. (C)21+4x2
    4. (D)22 ⁣(1+4x2)
  5. For the function f(x)=ln⁡(3x+5), which expression gives the derivative f′(x)?
    1. (A)3ln⁡(3x+5)
    2. (B)13
    3. (C)13x+5
    4. (D)33x+5
  6. The relation y3+3y=x3+2 defines y implicitly as a function of x. Which expression is dydx?
    1. (A)3x2−3y2
    2. (B)x2y2+1
    3. (C)y2+1x2
    4. (D)x2y2
  7. Differentiating f(x)=tan⁡(−2x+6) with respect to x gives f′(x)=
    1. (A)sec⁡2(−2)
    2. (B)−2sec⁡2(−2x+6)
    3. (C)2sec⁡2(−2x+6)
    4. (D)sec⁡2(−2x+6)
  8. The curve x3+xy−y2=−5 contains the point (1,−2). Which line is tangent to the curve at (1,−2)?
    1. (A)y+2=5(x−1)
    2. (B)y+2=−15(x−1)
    3. (C)y+2=15(x−1)
    4. (D)y+2=−35(x−1)
  9. The function f is differentiable everywhere, and y=[f(x2)]3. Which expression gives the value of dydx at x=3?
    1. (A)18f′(9)[f(9)]2
    2. (B)3f′(9)[f(9)]2
    3. (C)18[f(9)]2
    4. (D)18f′(3)[f(9)]2
  10. The slope of a curve satisfies dydx=y2−2x. Which expression gives d2ydx2 in terms of x and y?
    1. (A)−2+2y3−4xy
    2. (B)y2−2x
    3. (C)−2
    4. (D)2y3−4xy
  11. The function f is defined by f(x)=5x+11. At x=1, the derivative f′(1) equals
    1. (A)532
    2. (B)54
    3. (C)58
    4. (D)18
  12. The curve xy2−3x3y=4 passes through (1,−1). At that point, the tangent line to the curve has slope
    1. (A)−2
    2. (B)2
    3. (C)103
    4. (D)12
  13. Along the graph of y=(x3−cos⁡x)4, the slope dydx is given by which expression?
    1. (A)4 ⁣(x3−cos⁡x)3(3x2+sin⁡x)
    2. (B)4 ⁣(x3−cos⁡x)3
    3. (C)4 ⁣(x3−cos⁡x)3(3x2−sin⁡x)
    4. (D)4 ⁣(3x2+sin⁡x)3
  14. A curve in the xy-plane is described by 3x2+xy+y3=13. Which expression gives dydx in terms of x and y?
    1. (A)−3y2+6x+yx
    2. (B)−6x+y3y2
    3. (C)−6x+y3y2+x
    4. (D)−3y2+x6x+y
  15. The variables are linked by y=x2+2x and u=4x+3. Expressed in terms of x, the rate dydu is
    1. (A)x2+12
    2. (B)42x+2
    3. (C)8x+8
    4. (D)2x+2
  16. Differentiating the relation sin⁡(xy)=3x2 implicitly and solving for the slope gives dydx=
    1. (A)6xcos⁡(xy)
    2. (B)6x−yx
    3. (C)6x−yxcos⁡(xy)
    4. (D)6x−ycos⁡(xy)xcos⁡(xy)
  17. The differentiable functions f and g have the values and derivatives listed in the table. With h(x)=ln⁡(f(x)), what is h′(1)?
    x1234f(x)3123f′(x)25−3−6g(x)4133g′(x)−5662
    1. (A)−23
    2. (B)23
    3. (C)2
    4. (D)13
  18. The rule f(x)=23x3+4 defines a function. Its derivative f′(x) is
    1. (A)−2(3x3+4)2
    2. (B)−18x23x3+4
    3. (C)−18x2(3x3+4)2
    4. (D)29x2
  19. The graph of the function f, consisting of line segments, is shown. Let p(x)=f(x2+2x+5). What is p′(−2)?
    −4−3−2−101234567891011−4−3−2−101234xyy = f(x)
    1. (A)−4
    2. (B)5
    3. (C)1
    4. (D)−2

Part B: A graphing calculator is allowed.

Question 20
  1. Because f(x)=2x+arctan⁡x is increasing, it has an inverse function, which we call g. Find g′(−3).
    1. (A)2.459
    2. (B)−1.087
    3. (C)0.407
    4. (D)0.476

    End of Version A

Answer key · Version A

12345678910
DCDADBBBAA
3.13.33.13.43.13.23.13.23.13.6
11121314151617181920
CBACADBCDC
3.13.23.13.23.13.23.13.13.13.3

The second row is the correct letter; the third is the course topic each question tests.

Free AP Calculus AB worksheets at sigmaprep.io/worksheets/ap-calculus-abOriginal questions written by Sigma Prep. AP is a registered trademark of the College Board, which is not affiliated with this sheet.

What does the AP Calculus AB exam ask on Unit 3?

Less than a tenth of the multiple choice: Unit 3 was 27 of the 315 official multiple-choice questions on the seven 2012–2018 practice exams (8.6%), almost all of it without a calculator.

This worksheet has 20 questions in that mix: Part A (19) without a calculator and Part B (1) with one; the exam put 11% of its official Unit 3 multiple choice in the calculator section.

Counted from the official AP Calculus AB exams: multiple choice from 17 exams, 1969–2018, free response 2012–2026. No official question is reproduced here.

AP Calculus AB Unit 3 question types: how often the exam asks each

Official AP Calculus AB multiple-choice questions by question type, Unit 3 (Differentiation: Composite, Implicit, and Inverse Functions), and how many are on this worksheet
Question typeOn this worksheetOfficial questions, 1969–2018Exams it was on (of 17)On the 2012–2018 practice exams
Chain rule: trig, exponential and logTopic 3.1418146
Chain rule with composite or unspecified functionsTopic 3.14774
Chain rule: powers and rootsTopic 3.13773
Implicit differentiation for dy/dxTopic 3.2515146
Derivative of an inverse functionTopic 3.32996
Derivatives of inverse trig functionsTopic 3.41551
Higher-order derivativesTopic 3.61551

Counts are of official multiple-choice questions whose main topic is that kind, on 17 College Board exams 1969–2018 (7 released 1969–1998, 2003 and 2008 released, the 2008 practice exam and the seven 2012–2018 practice exams). No official multiple-choice questions after 2018 are public. Left out: choose a procedure for a derivative.

Free-response practice for this unit: 4 FRQs with a scoring guide.

Why these kinds of question: AP Calculus AB Unit 3 Review: The Most-Tested Chain Rule and Implicit Questions.

Questions about this worksheet

Unit 3, Differentiation: Composite, Implicit, and Inverse Functions, has 6 topics in the College Board's course description: the chain rule; implicit differentiation; the derivative of an inverse function; the derivatives of the inverse trig functions; choosing a procedure for a derivative; and higher-order derivatives. The course description weights the unit at 5–10% of the multiple-choice section.

Only in Part B, as on the exam. Questions 1–19 are Part A, with no calculator; question 20 is Part B, where the exam allows a graphing calculator (on the digital exam, the Desmos calculator built into Bluebook is available there). None of these Unit 3 questions needs one: Part B holds the one kind of question on this sheet that the exam has placed in its calculator section, the derivative of an inverse function. From May 2027 the exam's multiple-choice section is 29 questions without a calculator and 13 with one.

We tagged 760 official multiple-choice questions from 17 College Board exams (1969–2018) and the free-response questions from 1969 to 2026, sorted the Unit 3 questions into 8 kinds, and weighted each kind by how often and how recently the exam asks it. Every kind the exam actually uses gets at least one question, and the most-tested get more (5 on implicit differentiation for dy/dx, 4 on chain rule: trig, exponential and log, 4 on chain rule with composite or unspecified functions, 3 on chain rule: powers and roots, 2 on derivative of an inverse function). The one kind that has never been the main topic of an official multiple-choice question is left out.

The progress check is College Board's own end-of-unit quiz in AP Classroom, assigned by your teacher. Its questions are not public, and we do not have them: Sigma Prep is not affiliated with College Board. This worksheet is exam-style practice on the same unit: 20 original multiple-choice questions, weighted by how often the official exams ask each kind of question, with a no-calculator Part A and a calculator Part B. Time yourself, check the answer key, then redo the kinds you missed. For the free-response part, use the Unit 3 FRQ set, which has a scoring guide.

No. Every question here is original, written by Sigma Prep in the style of the exam. We studied which kinds of questions the official exams ask and how often; no official question is copied or reworded. Each answer was checked by an independent computer-algebra check before the sheet was published.

Yes. It is shown with the sheet, and the PDF prints it on its own last page. There is also a PDF without the key to hand out. Worked solutions are not included.

AP is a registered trademark of the College Board, which was not involved in making this worksheet and does not endorse it. All questions are original to Sigma Prep. Frozen 2026-10-24.