AP Calculus ABAP Exams

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

Sigma Prep

Math Instructor · 10+ Years Experience

October 24, 2026·8 min read

Key takeaways

Unit weight: 27 of 315 official multiple choice questions (8.6%) and 3.6% of the free-response points, almost all without a calculator.
Top types: implicit differentiation, and the chain rule on formulas and on values from a table or graph.
Practice: the chain rule and implicit differentiation by hand, until they are automatic.

Unit 3 is the smaller of the two units on computing derivatives: under a tenth of the AP Calculus AB multiple choice and under 4% of the free-response points. College Board’s Course and Exam Description (the CED) gives Unit 3 5–10% of the multiple choice. On the 7 official practice exams from 2012–2018 it was 27 of 315 questions (8.6%), between 3 and 6 per exam. On the 42-question section that starts in May 2027, that rate is about 3 or 4 questions.

In free response, Unit 3 earned 3.6% of the points on the 2012–2026 exams. 17 of the 86 free-response questions since 2012 had a Unit 3 part, and in 12 of them it was the main skill of the part. Two patterns cover most of them:

The chain rule with a table or graph (8 questions). A typical part: $h(x) = f(g(x))$ , the values of $f$ , $f'$ , $g$ and $g'$ are in a table, find $h'(a)$ . In 5 of the 8, the same question also had a product or quotient rule part, which is Unit 2.
Implicit differentiation as the opening step (5 questions). A whole question was built on an implicit curve in 2015, 2021, 2023, 2024 and 2025. In 2021, 2023 and 2025, part (a) was “show that $\frac{dy}{dx} = \ldots$ ”. The later parts, such as horizontal and vertical tangents and related rates, belong to Units 4 and 5 and carry most of the points.

The 5 most-tested Unit 3 question types

Ranked by share of the whole exam. The five add up to about 5.8% of the exam, out of 6.5% for all of Unit 3. Two smaller types follow them.

1. Implicit differentiation

Differentiate both sides with respect to $x$ , treating $y$ as a function of $x$ , then solve for $\frac{dy}{dx}$ .

Topic 3.21.5% of the exam5 of 7 practice exams (6 questions)5 FRQs since 2012, 4 of them since 2021

Try it

Find

\frac{dy}{dx}

for

x^3 + y^3 = 6xy

at the point

(3, 3)

Solution

$3x^2 + 3y^2 y' = 6y + 6xy'$ . At $(3, 3)$ that is $27 + 27y' = 18 + 18y'$ , so $y' = -1$ .

Common mistake

differentiating

xy

without the product rule, which drops the

x \cdot \frac{dy}{dx}

term or the

y

term, or forgetting the factor

\frac{dy}{dx}

on a

y

term.

Practice: Implicit Differentiation worksheet · AP Unit 3 worksheet

2. Chain rule with trig, exponential and log functions

Derivative of the outside, evaluated at the inside, times the derivative of the inside. Repeat for every layer.

Topic 3.11.4% of the exam4 of 7 practice exams (6 questions)3 FRQs since 2012

Try it

f(x) = \sin^2(3x)

. Find

f'\left(\frac{\pi}{12}\right)

Solution

$f'(x) = 2\sin(3x) \cdot \cos(3x) \cdot 3 = 3\sin(6x)$ , so $f'\left(\frac{\pi}{12}\right) = 3\sin\frac{\pi}{2} = 3$ .

Common mistake

leaving off the derivative of the inside function, or stopping one layer short in a nested function. Every layer contributes a factor.

Practice: The Chain Rule worksheet · AP Unit 3 worksheet

3. Chain rule with a table, a graph or unspecified functions

For $h(x) = f(g(x))$ , $h'(a) = f'(g(a)) \cdot g'(a)$ , with the values read from a table or graph.

Topic 3.11.4% of the exam4 of 7 practice exams8 FRQs since 2012, the most of any Unit 3 type

Try it

h(x) = f(g(x))

, with

g(2) = 5

g'(2) = -3

f'(2) = 7

and

f'(5) = 4

. Find

h'(2)

Solution

$f'(g(2)) \cdot g'(2) = f'(5) \cdot (-3) = -12$ .

Common mistake

evaluating

f'

x

instead of at

g(x)

(here

7 \cdot (-3) = -21

), or leaving off the factor

g'(x)

Practice: Derivatives From a Table worksheet · AP Unit 3 worksheet

4. Chain rule with powers and roots

Bring the power down, keep the inside, lower the power by 1, and multiply by the derivative of the inside.

Topic 3.10.8% of the exam3 of 7 practice exams1 FRQ since 2012

Try it

y = \sqrt{x^2 + 9}

. Find

\frac{dy}{dx}

x = 4

Solution

$\frac{dy}{dx} = \frac{x}{\sqrt{x^2 + 9}} = \frac{4}{5}$ .

Common mistake

leaving off the derivative of the inside: the derivative of

(2x^2 + 5)^7

7(2x^2 + 5)^6 \cdot 4x

, not

7(2x^2 + 5)^6

Practice: The Chain Rule worksheet · AP Unit 3 worksheet

5. Derivative of an inverse function

If $g$ is the inverse of $f$ , then $g'(a) = \frac{1}{f'(g(a))}$ . First find the $x$ where $f(x) = a$ .

Topic 3.30.7% of the exam6 of 7 practice exams, the most of any Unit 3 typeno FRQs since 2012 (the last was in 2007)

Try it

f(x) = x^3 + 2x + 1

, and

g

is the inverse of

f

. Find

g'(4)

Solution

$f(1) = 4$ , so $g(4) = 1$ and $g'(4) = \frac{1}{f'(1)} = \frac{1}{5}$ .

Common mistake

evaluating

f'

a

itself. Here

\frac{1}{f'(4)} = \frac{1}{50}

is the trap answer.

Practice: Derivative of an Inverse Function worksheet · AP Unit 3 worksheet

Two more Unit 3 types, lower priority

6. Higher-order derivatives

Differentiate again. Each new derivative of a composite brings in another factor of the inside’s derivative.

Topic 3.60.4% of the exam1 of 7 practice examsno FRQs since 2012 as its own type

Try it

y = \cos(3x)

. Find

\frac{d^2y}{dx^2}

Solution

$\frac{dy}{dx} = -3\sin(3x)$ , so $\frac{d^2y}{dx^2} = -9\cos(3x)$ .

Common mistake

multiplying by the inside’s derivative only once, which gives

-3\cos(3x)

The free-response version is filed under Unit 5: find $\frac{d^2y}{dx^2}$ from a $\frac{dy}{dx}$ written in terms of $x$ and $y$ , usually from a differential equation. That was in 7 FRQs since 2012.

Practice: Higher Order Derivatives worksheet · AP Unit 3 worksheet

7. Derivatives of inverse trig functions

Know the forms for arcsin and arctan, and multiply by the derivative of the inside.

Topic 3.40.3% of the exam1 of 7 practice exams1 FRQ since 2012 (2025, inside a limit question)

Try it

y = \arctan(2x)

. Find

\frac{dy}{dx}

x = \frac{1}{2}

Solution

$\frac{2}{1 + 4x^2} = \frac{2}{2} = 1$ .

Common mistake

mixing up the forms. The derivative of

\arcsin u

\frac{u'}{\sqrt{1 - u^2}}

, but the derivative of

\arctan u

\frac{u'}{1 + u^2}

. The other common slip is forgetting the factor

u'

Practice: Derivatives of Inverse Trig Functions worksheet · AP Unit 3 worksheet

What’s rising and fading in Unit 3

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. No Unit 3 type is Rising. The counts here are small, so read every line as a direction.

Steady, but up: the chain rule with a table, graph or unspecified functions. 1.66% of recent exams against 0.97% across all years, short of double.
Steady, but down: the chain rule on trig, exponential and log formulas. 1.21% against 1.72%. Its latest free-response use was in 2017.
Steady: the derivative of an inverse function. 0.95% against 0.65%, and on 6 of the 7 practice exams.
Fading: derivatives of inverse trig functions. 0.16% against 0.35%. That rests on 5 official multiple choice questions in all, with one on the 2012–2018 practice exams (2016).
Fading, but still drilled: higher-order derivatives. 0.16% against 0.41%, with one question on the 2012–2018 practice exams (2018). Teacher review material written since 2019 still uses it heavily.

The newest free response, read with care. Comparing the free-response points from 2019–2026 with those from 2012–2018, implicit differentiation went from 0.0% to 1.6%, while the table-and-graph chain rule went from 2.9% to 1.2% and the chain rule on formulas from 1.1% to 0.0%. Unit 3 as a whole went from 4.5% to 2.8%. One question is about 2.3% of a window’s points, so each of these moves is a question or less. The direction is consistent, though: the newest free response leans less on the chain rule and more on implicit curves.

In the CED, rarely or never tested

Selecting procedures for calculating derivatives (topic 3.5). No official question has it as its main topic. The CED frames it as a skill topic: choosing the right rule, which every derivative question needs.
Inverse function derivatives on free response. On the multiple choice they are a regular (6 of 7 practice exams), but the last free-response part was in 2007.
Inverse trig derivatives and higher-order derivatives, covered above: each had a single question on the 2012–2018 practice exams.

What students over- and under-practice in Unit 3

We tagged 10,263 multiple choice questions from teacher worksheets, review packets and prep books (reposted official questions and non-AB content removed). Unit 3 is 10.8% of them, against 9.2% of the 719 in-scope official multiple choice questions from 1969–2018. Per type:

Question type	Practice vs official share	Ratio
Higher-order derivatives	1.61% vs 0.70%	2.3× over
Chain rule with a table, graph or unspecified functions	1.50% vs 0.97%	1.5× over
Implicit differentiation	2.68% vs 2.09%	1.3× over
Chain rule with trig, exponential and log	2.54% vs 2.50%	1.0×, even
Chain rule with powers and roots	0.98% vs 0.97%	1.0×, even
Derivatives of inverse trig functions	0.57% vs 0.70%	0.8× under
Derivative of an inverse function	0.95% vs 1.25%	0.8× under

Official counts per type are small (5 to 18 questions), so read the ratios as directions.

Higher-order derivatives are the clear over-practice: more than twice their exam share, on a type with one question on the 2012–2018 practice exams.
The chain rule on formulas is not over-drilled. It is practiced at about its official share. The extra chain rule practice goes to the table-and-graph version, which is also the version the free response uses most.
Inverse function derivatives get too little practice for a type that was on 6 of the 7 practice exams.
The free-response format is the gap. Implicit differentiation is practiced a bit above its share in multiple choice, but the free response asks you to show that $\frac{dy}{dx}$ equals a given expression, and then to use it. Multiple choice practice never asks you to write that work.

Calculator note

Unit 3 is almost all no-calculator. Of the 27 Unit 3 multiple choice questions on the 2012–2018 practice exams, 24 were in the no-calculator part. The 3 in the calculator part were 2 inverse function derivatives, both given as a table of values, and 1 chain rule question on trig, exponential and log functions. On free response, 14 of the 17 Unit 3 questions were no-calculator questions, including all 5 implicit-curve questions. The exceptions were 2015 Question 2, the 2020 at-home exam (calculator allowed throughout) and 2025 Question 1, where the inverse trig derivative came from a model.

So practice the chain rule and implicit differentiation by hand until they are automatic.

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 3

Our free AP Calculus AB Unit 3 worksheet has 20 original multiple choice questions on the chain rule, implicit differentiation and inverse functions, 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 3 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 3 Progress Check in AP Classroom.

Common Questions

The CED gives Unit 3 5–10% of the multiple choice. On the 7 official practice exams from 2012–2018 it was 27 of 315 multiple choice questions (8.6%), 3 to 6 per exam. In free response it earned 3.6% of the points on the 2012–2026 exams, mostly as single parts: the chain rule with a table or graph, and implicit differentiation at the start of an implicit-curve question.

Usually as the opening part of a question built on an implicit curve: show that dy/dx equals a given expression, then use it to find tangent lines, horizontal or vertical tangents, or a related rate. Such a question appeared in 2015, 2021, 2023, 2024 and 2025. In multiple choice, implicit differentiation was on 5 of the 7 official practice exams from 2012–2018.

If g is the inverse of f, then g'(a) = 1/f'(g(a)). Find the x where f(x) = a first, then take the reciprocal of f' at that x, not at a. This type was on 6 of the 7 official practice exams from 2012–2018, the most of any Unit 3 type.

Yes, they are in the CED (topic 3.4), but rarely. There are 5 official multiple choice questions in all, one of them on the 2012–2018 practice exams, and one free-response part since 2012 (2025) used an arctan derivative inside a limit. Know the arcsin and arctan forms, and give them less time than the chain rule and implicit differentiation.

Almost never. Of the 27 Unit 3 multiple choice questions on the 2012–2018 practice exams, 24 were in the no-calculator part, and 14 of the 17 free-response questions with a Unit 3 part since 2012 were no-calculator questions, including every implicit-curve question.

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

The 5 most-tested Unit 3 question types

1. Implicit differentiation

2. Chain rule with trig, exponential and log functions

3. Chain rule with a table, a graph or unspecified functions

4. Chain rule with powers and roots

5. Derivative of an inverse function

Two more Unit 3 types, lower priority

6. Higher-order derivatives

7. Derivatives of inverse trig functions

What’s rising and fading in Unit 3

In the CED, rarely or never tested

What students over- and under-practice in Unit 3

Calculator note

How we counted

Practice Unit 3

Common Questions

How much of the AP Calculus AB exam is Unit 3?+

How is implicit differentiation tested on the AP Calculus AB free response?+

How do you find the derivative of an inverse function?+

Are inverse trig derivatives on the AP Calculus AB exam?+

Do I need a calculator for Unit 3 questions?+

Practice this unit: free AP Calculus AB Unit 3 worksheet

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