Calc AB · Unit 3 Progress Check

AP Calculus AB Unit 3 Progress Check Differentiation: Composite, Implicit, and Inverse Functions walkthrough.

AP Calculus AB Unit 3 is the chain rule unit, and the Unit 3 Progress Check — both the MCQ and the FRQ Part A — is where differentiation stops being mechanical. This walkthrough covers what the questions test, the traps, and how to keep composite derivatives straight. Explanations only — no AP Classroom answer keys.

Updated August 2026Written by Mahmudul HasanFree · No signup

What Unit 3 covers

Unit 3 is Differentiation: Composite, Implicit, and Inverse Functions. Roughly 9–13% of the AP Calculus AB exam, and a prerequisite for everything that follows.

The chain rule

d/dx[f(g(x))] = f′(g(x))·g′(x) — the rule the rest of the course depends on.

Implicit differentiation

Differentiating both sides with respect to x when y is not isolated, remembering dy/dx.

Derivatives of inverse functions

Using the relationship between a function and its inverse at corresponding points.

Inverse trigonometric derivatives

arcsin, arccos, and arctan.

Higher-order derivatives

Second derivatives and what they describe.

The Progress Check MCQ: what each question type tests

These are the question patterns that recur on this Progress Check, and what each one is really asking.

Nested chain rule

Two or three layers deep. Work outside in and write each layer before simplifying.

Implicit differentiation

Every y term produces a dy/dx. Collect and solve for it.

Inverse function derivatives

Given f and a point, find the derivative of f⁻¹. Swap the coordinates and take the reciprocal.

Table-based composite derivatives

Values of f, g, f′, g′ in a table, asking for the derivative of the composite at a point. Pure chain-rule bookkeeping.

Related-rate setup

Beginning to apply the chain rule to changing quantities.

The Progress Check FRQ

Unit 3 free response often asks for an implicit derivative, a tangent line to an implicitly defined curve, or a composite derivative from a table. Points come from correct chain-rule application, clear notation showing dy/dx, and solving for the requested quantity rather than stopping halfway.

Where students lose the most points

Forgetting the inner derivative

The single most common calculus error. If you differentiate sin(3x) to cos(3x), you have lost the 3.

Dropping dy/dx in implicit differentiation

Every y differentiated with respect to x carries dy/dx.

Mishandling products inside composites

Products and chains often appear together; apply both rules deliberately.

Inverting the inverse-derivative relationship

It is the reciprocal of the derivative at the corresponding point, not at the same x.

Simplifying too early

Write every layer first. Cleaning up mid-derivative is where errors hide.

How to work through this unit

Write composite derivatives in layers before simplifying: outer derivative, then inner, then multiply. It is slower for one line and far faster overall because you stop losing the inner factor.

Once you have finished the Progress Check, put your raw score into our AP Calculus AB Calculator to see roughly where that pace puts you on the 1–5 scale, and use the AP Calculus AB Review for the full exam format and study plan.

Frequently asked questions

Quick answers — written by humans, not a chatbot.

What does the AP Calculus AB Unit 3 Progress Check cover?

The chain rule, implicit differentiation, derivatives of inverse and inverse trigonometric functions, and higher-order derivatives.

What is the most common Unit 3 mistake?

Forgetting the inner derivative when applying the chain rule. Writing the derivative in layers before simplifying prevents it.

How do I find the derivative of an inverse function?

Use the corresponding point: the derivative of the inverse at a point equals the reciprocal of the original function’s derivative at the matching point.

Do you post Unit 3 Progress Check FRQ answers for AP Calculus?

No. We publish walkthroughs of the reasoning and the traps rather than AP Classroom answer keys.