Worksheets · AP Calculus AB

AP Calculus AB Unit 2 Worksheet: Derivatives

20 exam-style multiple-choice questions on the definition of the derivative and the basic derivative rules, free to use in class. Good practice before the Unit 2 Progress Check in AP Classroom. Free response for this unit: Unit 2 FRQ set.

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AP Calculus AB Unit 2: Derivatives

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 1–15
  1. The limit lim⁡x→2x4−16x−2 is equal to which of the following?
    1. (A)32
    2. (B)16
    3. (C)8
    4. (D)nonexistent
  2. With constants a and b, a function is given in two pieces: f(x)={ax3+bx,x≤12x2+3,x>1. For which choice of a and b is f differentiable at x=1?
    1. (A)a=−12, with b free to take any value
    2. (B)a=−12 and b=112
    3. (C)a=112 and b=−12
    4. (D)a=12 and b=−112
  3. Over the interval 2≤x≤5, find the average rate of change of g(x)=6x.
    1. (A)35
    2. (B)−35
    3. (C)2110
    4. (D)−625
  4. For the function f(x)=3x−2⋅3x+4, find the exact value of the derivative f′(0).
    1. (A)−2ln⁡3
    2. (B)3−2ln⁡3
    3. (C)3
    4. (D)3−2ln⁡3
  5. Suppose that f(x)=5csc⁡x+x. Written in terms of x, the derivative f′(x) is
    1. (A)5csc⁡(x)cot⁡x+1
    2. (B)1−5csc⁡(x)cot⁡x
    3. (C)1−5csc⁡2x
    4. (D)5csc⁡(x)cot⁡x−1
  6. Suppose f is differentiable for all real numbers. Which of the limits below must equal f′(1)?
    I. lim⁡x→1f(x+h)−f(x)h
    II. lim⁡h→0f(1+h)−f(1)h
    III. lim⁡x→1f(x)−f(1)x−1
    1. (A)II and III only
    2. (B)I and III only
    3. (C)III only
    4. (D)II only
  7. Consider the curve y=x2−x+3. Which equation describes the line tangent to this curve at the point where x=−1?
    1. (A)y=−3x+5
    2. (B)y=x3+163
    3. (C)y=−3x−8
    4. (D)y=−3x+2
  8. A curve in the plane is the graph of y=−x2cos⁡x. Which expression gives dydx, the slope of the curve at each of its points?
    1. (A)2xsin⁡x
    2. (B)−2xcos⁡x−x2sin⁡x
    3. (C)x2sin⁡x−2xcos⁡x
    4. (D)−2xcos⁡x
  9. At x=π2, the graph of y=4cos⁡x has a tangent line. Which of the following is that line?
    1. (A)y=−4x+2π
    2. (B)y=x4−π8
    3. (C)y=0
    4. (D)y=−4x
  10. The function f is given by f(x)=−2x2+5x3. At x=−3, how fast is f(x) changing with respect to x?
    1. (A)4013
    2. (B)−12179
    3. (C)364127
    4. (D)−36419
  11. The function f is defined by the rule f(x)=4ln⁡xx+1. Which expression below equals f′(x)?
    1. (A)4(x+1)x+4ln⁡x(x+1)2
    2. (B)4(x+1)x−4ln⁡x(x+1)2
    3. (C)4(x+1)x−4ln⁡xx+1
    4. (D)4x
  12. The line x−2y=1 is drawn in the same plane as the curve y=x22−3x. Find the point of the curve at which the tangent line has the same slope as this line.
    1. (A)(72,−358)
    2. (B)(1,−52)
    3. (C)(5,−52)
    4. (D)(52,−358)
  13. A function f is built from two pieces: f(x)={6x−4,x<12x2,x≥1. Which of the statements below about f at x=1 are true?
    I. lim⁡x→1f(x) exists.
    II. The two pieces of f join without a break at x=1.
    III. f′(1) exists.
    1. (A)I and II only
    2. (B)I only
    3. (C)II only
    4. (D)I, II, and III
  14. Use the two segment graphs shown, of f and of g, to find the derivative of h(x)=f(x)g(x) at x=1.
    −10123456789−3−2−1012345xyy = f(x)−10123456789−2−101234xyy = g(x)
    1. (A)−14
    2. (B)12
    3. (C)−1
    4. (D)14
  15. The figure shows the graph of a function f, with the points at x=1,5,9,13 marked. Where, among the marked x-values, does the graph of f have no break but also no derivative?
    −10123456789101112131415−1012345678xyy = f(x)
    1. (A)x=13
    2. (B)x=1
    3. (C)x=5
    4. (D)x=9

Part B: A graphing calculator is allowed.

Questions 16–20
  1. For the differentiable functions f and g, the table lists their values and the values of their derivatives at several inputs. Let h(x)=f(x)g(x). Find h′(2).
    x123f(x)−4−4−6f′(x)−73−2g(x)211g′(x)1−74
    1. (A)−25
    2. (B)−21
    3. (C)31
    4. (D)3
  2. The figure shows the graph of y=f(x), where f is differentiable; three points of the graph, at x=0, x=2 and x=4, are marked. Which ordering of the slopes of the graph at the marked points is correct?
    −6−5−4−3−2−10123456789−6−5−4−3−2−10123456xyy = f(x)
    1. (A)f′(2)<f′(0)<f′(4)
    2. (B)f′(4)<f′(0)<f′(2)
    3. (C)f′(0)<f′(2)<f′(4)
    4. (D)f′(4)<f′(2)<f′(0)
  3. Let f(x)=xcos⁡x. The graph of f has a tangent line at the point with x-coordinate 2.6, and this line passes through (0,b). What is b?
    1. (A)0.809
    2. (B)2.495
    3. (C)−4.234
    4. (D)1.470
  4. Let f(x)=exx. What is the value of f′(2.2)?
    1. (A)6.085
    2. (B)26.773
    3. (C)−4.702
    4. (D)4.702
  5. Let f(x)=(x2+1)sin⁡x. For exactly one x with 0.3<x<2.5, the graph of f has a tangent line parallel to y=−2.5x−2. At which x does this happen?
    1. (A)2.165
    2. (B)2.214
    3. (C)2.471
    4. (D)2.425

    End of Version A

Answer key · Version A

12345678910
ABBBBADCAC
2.12.42.12.72.102.12.22.82.22.5
11121314151617181920
BAADCCDDDC
2.92.22.42.82.42.82.32.22.32.2

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 2?

About a seventh of the multiple choice: Unit 2 was 44 of the 315 official multiple-choice questions on the seven 2012–2018 practice exams (14.0%).

This worksheet has 20 questions in that mix: Part A (15) without a calculator and Part B (5) with one; the exam put 26% of its official Unit 2 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 2 question types: how often the exam asks each

Official AP Calculus AB multiple-choice questions by question type, Unit 2 (Differentiation: Definition and Fundamental Properties), 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
Recognize a limit as a derivativeTopic 2.1214137
Average rate of changeTopic 2.11544
Equation of a tangent lineTopic 2.2315124
Find where the tangent has a given slopeTopic 2.22862
Estimate or compare derivatives from a table or graphTopic 2.32443
Is f differentiable at a point?Topic 2.4213114
Constants that make a piecewise function differentiableTopic 2.41221
Power rule derivativesTopic 2.51975
Derivatives of sin, cos, eˣ, ln x (and aˣ)Topic 2.711072
Product or quotient rule from a table or graphTopic 2.82756
Product ruleTopic 2.8110104
Quotient ruleTopic 2.91882
Derivatives of tan, sec, cot, cscTopic 2.101220

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: constant, sum, difference and constant multiple rules.

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

Why these kinds of question: AP Calculus AB Unit 2 Review: The Most-Tested Derivative Questions.

Questions about this worksheet

Unit 2, Differentiation: Definition and Fundamental Properties, has 10 topics in the College Board's course description: average and instantaneous rates of change; the derivative as a limit and derivative notation, including tangent lines; estimating a derivative from a table or graph; where a function is and is not differentiable; the power rule; the constant, sum and difference rules; the derivatives of sin x, cos x, eˣ and ln x; the product and quotient rules; and the derivatives of tan x, cot x, sec x and csc x. The course description weights the unit at 10–15% of the multiple-choice section.

Only in Part B, as on the exam. Questions 1–15 are Part A, with no calculator; questions 16–20 are 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 2 questions needs one: Part B holds the kinds of questions the exam has placed in its calculator section, such as estimating a derivative from a table or a graph and the product rule with values from a table. 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 2 questions into 14 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 (3 on equation of a tangent line, 2 on product or quotient rule from a table or graph, 2 on recognize a limit as a derivative, 2 on estimate or compare derivatives from a table or graph, 2 on find where the tangent has a given slope, 2 on is f differentiable at a point?). 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 2 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.