AP Calculus ABAP Exams

AP Calculus AB Unit 2 Review: The Most-Tested Derivative Questions

Sigma Prep

Math Instructor · 10+ Years Experience

October 24, 2026·9 min read

Key takeaways

Unit weight: 44 of 315 official multiple choice questions (14.0%) and 8.0% of the free-response points, usually one part of a larger question.
Top types: the equation of a tangent line, the product or quotient rule from a table or graph, and a limit read as a derivative.
Practice: estimating a derivative from a table, with units. It was on the free response of 12 of the 15 exams since 2012.

Unit 2 is about a seventh of the AP Calculus AB multiple choice and about a twelfth of the free-response points. College Board’s Course and Exam Description (the CED) gives Unit 2 10–15% of the multiple choice. On the 7 official practice exams from 2012–2018 it was 44 of 315 questions (14.0%), between 4 and 8 per exam, near the top of that range. On the 42-question section that starts in May 2027, that rate is about 6 questions.

In free response, Unit 2 earned 8.0% of the points on the 2012–2026 exams. It rarely gets a question of its own, but 35 of the 86 free-response questions since 2012 had a Unit 2 part. The common ones:

Estimate a derivative from a table (12 questions, on 12 of the 15 exams). In 11 of them it was part (a) of a question built on a table, and on 6 exams it was the first thing asked: Question 1(a).
Write a tangent line (13 questions). In 7 of them the line was then used for a tangent-line approximation, which is a Unit 4 topic.
Product or quotient rule with values from a table or graph (7 questions).
Average rate of change (5) and differentiability at a point (2).

The 10 most-tested Unit 2 question types

Ranked by share of the whole exam. The ten add up to about 10.7% of the exam, nearly all of Unit 2’s 11.6%.

1. Equation of a tangent line

The point comes from $f(a)$ , the slope from $f'(a)$ : $y = f(a) + f'(a)(x - a)$ .

Topic 2.21.7% of the exam4 of 7 practice exams13 FRQs since 2012, 7 of them as the line for a tangent-line approximation

Try it

Give the tangent line to the curve

y = \ln x + x^2

where

x = 1

Solution

$f(1) = 1$ and $f'(1) = 1 + 2 = 3$ , so $y - 1 = 3(x - 1)$ .

Common mistake

using

f(a)

as the slope instead of

f'(a)

Practice: Tangent and Normal Lines worksheet · AP Unit 2 worksheet

2. Product or quotient rule from a table or graph

The same rules, with the values read from a table or a graph instead of formulas.

Topic 2.81.5% of the exam4 of 7 practice exams (6 questions)7 FRQs since 2012

Try it

f(1) = 4

f'(1) = -2

g(1) = 2

g'(1) = 3

, and

h(x) = \frac{f(x)}{g(x)}

. Find

h'(1)

Solution

$\frac{f'g - fg'}{g^2} = \frac{(-2)(2) - (4)(3)}{2^2} = \frac{-16}{4} = -4$ .

Common mistake

multiplying or dividing the derivatives (here

-\frac{2}{3}

), or reversing the order in the quotient-rule numerator.

Practice: Derivatives From a Table worksheet · AP Unit 2 worksheet

3. Recognize a limit as a derivative

A limit of the form $\frac{f(a + h) - f(a)}{h}$ is $f'(a)$ . Name $f$ and $a$ , then differentiate.

Topic 2.11.4% of the examall 7 practice exams, the only Unit 2 type on every oneno FRQs since 2012

Try it

Find

\lim\limits_{h \to 0} \frac{\cos\left(\frac{\pi}{3} + h\right) - \frac{1}{2}}{h}

Solution

it is $f'\left(\frac{\pi}{3}\right)$ for $f(x) = \cos x$ , so $-\sin\frac{\pi}{3} = -\frac{\sqrt{3}}{2}$ .

Common mistake

getting

\frac{0}{0}

and answering “does not exist” instead of recognizing the definition of

f'(a)

Practice: Limits in the Form of a Derivative worksheet · AP Unit 2 worksheet

4. Estimate a derivative from a table or graph

Use the average rate of change over the table interval around the point. On free response, show the computation and give units.

Topic 2.31.3% of the exam3 of 7 practice exams12 FRQs since 2012, the most of any Unit 2 type

Try it

A tank holds

W(t)

gallons at time

t

hours. Selected values are in the table.

$t$ (hours)	0	3	5	9
$W(t)$ (gallons)	120	135	150	162

Estimate

W'(4)

Solution

$\frac{150 - 135}{5 - 3} = 7.5$ gallons per hour.

Common mistake

using an interval that does not surround the point, such as

[0, 3]

, instead of the one centered on it.

Practice: Instantaneous Rate of Change worksheet · AP Unit 2 worksheet

5. Is $f$ differentiable at a point?

Differentiable needs continuity and matching slopes from both sides. Corners, cusps, vertical tangents and breaks in the graph all fail.

Topic 2.41.0% of the exam4 of 7 practice exams2 FRQs since 2012

Try it

f(x) = x^2 + 1

for

x < 1

and

2x + 1

for

x \ge 1

. Is

f

differentiable at

x = 1

Solution

no. Both pieces have slope 2 at $x = 1$ , but the left piece approaches 2 while $f(1) = 3$ , so $f$ is not even continuous there.

Common mistake

calling

f

differentiable because it is continuous, or because the slopes of the two pieces match.

Practice: Differentiability worksheet · AP Unit 2 worksheet

6. Find where the tangent has a given slope

Set $f'(x)$ equal to the slope you want and solve. Parallel lines have equal slopes.

Topic 2.21.0% of the exam2 of 7 practice exams1 FRQ since 2012 (2025)

Try it

At what point is the tangent to

y = x^2 - 4x

parallel to the line

2x + y = 5

Solution

the line has slope $-2$ , and $2x - 4 = -2$ at $x = 1$ , so the point is $(1, -3)$ .

Common mistake

reading the slope of

ax + by = c

a

\frac{a}{b}

. Solve for

y

first: the slope is

-\frac{a}{b}

Practice: Tangent and Normal Lines worksheet · AP Unit 2 worksheet

7. Product rule

$(uv)' = u'v + uv'$ .

Topic 2.80.9% of the exam4 of 7 practice exams1 FRQ since 2012

Try it

y = x^3 \ln x

. Find

\frac{dy}{dx}

x = e

Solution

$3x^2 \ln x + x^2 = 3e^2 + e^2 = 4e^2$ .

Common mistake

multiplying the derivatives:

(uv)'

is not

u'v'

Practice: The Product Rule worksheet · AP Unit 2 worksheet

8. Power rule

Rewrite roots and reciprocals as powers, then bring the exponent down and lower it by 1.

Topic 2.50.8% of the exam4 of 7 practice examsno FRQs since 2012

Try it

f(x) = 4\sqrt{x} - \frac{6}{x^2}

. Find

f'(4)

Solution

$f'(x) = 2x^{-1/2} + 12x^{-3}$ , so $f'(4) = 1 + \frac{12}{64} = \frac{19}{16}$ .

Common mistake

raising a negative exponent toward 0 instead of lowering it: the derivative of

x^{-2}

-2x^{-3}

, not

-2x^{-1}

Practice: The Power Rule worksheet · AP Unit 2 worksheet

9. Average rate of change

$\frac{f(b) - f(a)}{b - a}$ : the slope of the secant line.

Topic 2.10.7% of the exam3 of 7 practice exams5 FRQs since 2012

Try it

Find the average rate of change of

f(x) = x^3 - x

[1, 3]

Solution

$\frac{24 - 0}{3 - 1} = 12$ .

Common mistake

giving the change

f(b) - f(a)

without dividing by

b - a

, or giving the instantaneous rate at an endpoint.

Practice: Average Rate of Change worksheet · AP Unit 2 worksheet

10. Quotient rule

$\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}$ .

Topic 2.90.6% of the exam2 of 7 practice examsno FRQs since 2012

Try it

y = \frac{x^2 + 1}{x - 3}

. Find

\frac{dy}{dx}

x = 1

Solution

$\frac{2x(x - 3) - (x^2 + 1)}{(x - 3)^2} = \frac{-4 - 2}{4} = -\frac{3}{2}$ .

Common mistake

reversing the numerator.

uv' - u'v

gives the right size with the wrong sign.

Practice: The Quotient Rule worksheet · AP Unit 2 worksheet

What’s rising and fading in Unit 2

We compare each type’s share of recent exams (multiple choice 2012–2018, free response 2012–2026) with its share across all years: Rising is at least double, Fading under half. The two risers are both about reading derivatives from tables and graphs.

Rising: the product or quotient rule from a table or graph. 2.26% of recent exams against 0.85% across all years. 6 of its 7 official multiple choice questions are from 2012–2018, and it was in 7 FRQs since 2012. A small base, but it all points one way.
Rising: estimating a derivative from a table or graph. 1.75% against 0.82%, driven by free response. It has only 4 official multiple choice questions in all (3 from 2012–2018), but it was in a free-response question on 12 of the 15 exams from 2012 to 2026.
Fading: the equation of a tangent line. 1.26% against 3.16%. Its free-response points fell from 1.9% (2012–2018) to 0.7% (2019–2026), with the latest part in 2022. It still ranks first in Unit 2 because every linear approximation needs one.
Fading: finding where the tangent has a given slope (0.45% against 1.18%) and the quotient rule (0.32% against 0.74%). Each had 2 questions on the 2012–2018 practice exams.

All three are still common in teacher review material written since 2019. Unit 2’s share of free-response points went from 8.5% (2012–2018) to 7.5% (2019–2026). One question is about 2.3% of a window’s points, so that is less than half a question.

In the CED, rarely or never tested

Derivatives of tan, sec, cot and csc (topic 2.10). Two official multiple choice questions ever, the last in 1998. You still need them inside chain rule questions.
Constants that make a piecewise function differentiable (topic 2.4). Two official multiple choice questions ever (the latest in 2013) and no free-response part since 2003.
Derivatives of $\sin x$ , $\cos x$ , $e^x$ and $\ln x$ on their own (topic 2.7). 10 official multiple choice questions, only 2 of them from 2012–2018 (the latest in 2014), and one free-response part, 2026 Question 4(a). Almost every other derivative question uses them.
The sum, difference and constant multiple rules (topic 2.6). No official question has them as its main topic; every derivative question uses them.

What students over- and under-practice in Unit 2

We tagged 10,263 multiple choice questions from teacher worksheets, review packets and prep books (reposted official questions and non-AB content removed). Unit 2 is 17.2% of them, against 14.9% of the 719 in-scope official multiple choice questions from 1969–2018. Per type:

Question type	Practice vs official share	Ratio
Constants that make a piecewise function differentiable	0.48% vs 0.28%	1.7× over
Equation of a tangent line	3.12% vs 2.09%	1.5× over
Estimate a derivative from a table or graph	0.83% vs 0.56%	1.5× over
Find where the tangent has a given slope	1.63% vs 1.11%	1.5× over
Average rate of change	0.94% vs 0.70%	1.3× over
Product rule	1.75% vs 1.39%	1.3× over
Quotient rule	1.33% vs 1.11%	1.2× over
Recognize a limit as a derivative	2.25% vs 1.95%	1.2× over
Product or quotient rule from a table or graph	1.04% vs 0.97%	1.1×, even
Is f differentiable at a point?	1.73% vs 1.81%	1.0×, even
Power rule	1.16% vs 1.25%	0.9×, even
Derivatives of sin, cos, $e^x$ and $\ln x$	0.54% vs 1.39%	0.4× under

Official counts per type are small (2 to 15 questions), so read the ratios as directions.

Unit 2 practice is close to the multiple choice. No type is above 1.7×, and that one rests on 2 official questions.
Tangent lines are the biggest gap in points (about 1 percentage point), on a type that is fading.
The table estimate is a free-response skill. The exam asks it mostly in free response, 12 questions since 2012, usually with units and sometimes an interpretation in context. Practice writing that answer, not only picking it.
The one under-practiced type matters mostly on old exams: only 2 of its 10 official questions are from 2012–2018.

Calculator note

Of the 44 Unit 2 multiple choice questions on the 2012–2018 practice exams, 33 were in the no-calculator part and 11 in the calculator part. 8 of those 11 gave the function as a table or a graph, including all 3 derivative estimates and 3 of the 6 product or quotient rule questions with values. None of the limit-as-derivative, power rule, product rule or quotient rule questions on formulas was in the calculator part. On free response, 6 of the 12 table estimates were in calculator questions (counting 2020, when a calculator was allowed throughout), though the estimate itself is arithmetic.

So take derivatives by hand, and expect table and graph questions in either part.

How we counted

Every number here comes from the public official AP Calculus AB exams: 760 multiple choice questions (1969–2018) and 416 free-response questions (1969–2026), each tagged to its CED topic and question type. Every example problem is our own.

Practice Unit 2

Our free AP Calculus AB Unit 2 worksheet has 20 original multiple choice questions, a no-calculator Part A and a calculator Part B, weighted by how often the official exams ask each type. It comes with an answer key and a printable PDF.

For free response, the AP Calculus AB Unit 2 FRQ set has 4 multi-part questions in the exam’s form, each scored out of 9 points, with a scoring guide and a PDF. Both make good practice before the Unit 2 Progress Check in AP Classroom.

Common Questions

The CED gives Unit 2 10–15% of the multiple choice. On the 7 official practice exams from 2012–2018 it was 44 of 315 multiple choice questions (14.0%), 4 to 8 per exam. In free response it earned 8.0% of the points on the 2012–2026 exams, mostly as single parts: estimating a derivative from a table, writing a tangent line, and the product or quotient rule with values from a table or graph.

Yes, but less than before. Writing a tangent line is 1.26% of recent exams against 3.16% across all years, so it counts as fading. It was still in 13 free-response questions from 2012 to 2026, 7 of them as the line for a tangent-line approximation, and it is the top Unit 2 type in our ranking.

Take the two table values on either side of the point (or the interval the question names), divide the change in the function by the change in the input, and give units: the function's units per unit of the input. Show the computation. This was asked in 12 free-response questions from 2012 to 2026, on 12 of the 15 exams.

Not necessarily. f(x) = |x| is continuous at 0 but has a corner there, so it has no derivative at 0. The arrow only goes one way: differentiable implies continuous. To show a piecewise function is differentiable at the break, check that the pieces meet and that their slopes match.

Mostly not. Of the 44 Unit 2 multiple choice questions on the 2012–2018 practice exams, 33 were in the no-calculator part. 8 of the 11 in the calculator part gave the function as a table or graph. The power, product and quotient rule questions on formulas were all in the no-calculator part.

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AP Calculus AB Unit 2 Review: The Most-Tested Derivative Questions

The 10 most-tested Unit 2 question types

1. Equation of a tangent line

2. Product or quotient rule from a table or graph

3. Recognize a limit as a derivative

4. Estimate a derivative from a table or graph

5. Is fff differentiable at a point?

6. Find where the tangent has a given slope

7. Product rule

8. Power rule

9. Average rate of change

10. Quotient rule

What’s rising and fading in Unit 2

In the CED, rarely or never tested

What students over- and under-practice in Unit 2

Calculator note

How we counted

Practice Unit 2

Common Questions

How much of the AP Calculus AB exam is Unit 2?+

Are tangent line questions still on the AP Calculus AB exam?+

How do you estimate a derivative from a table on the AP Calculus free response?+

If a function is continuous, is it differentiable?+

Do I need a calculator for Unit 2 questions?+

Practice this unit: free AP Calculus AB Unit 2 worksheet

Keep reading

AP Calculus AB Review: The Most-Tested Questions, Unit by Unit

AP Calculus AB Unit 1 Review: The Most-Tested Limits and Continuity Questions

AP Calculus AB Unit 3 Review: The Most-Tested Chain Rule and Implicit Questions

5. Is $f$ differentiable at a point?