Calc AB · Unit 2 Progress Check

AP Calculus AB Unit 2 Progress Check Differentiation: Definition and Fundamental Properties walkthrough.

AP Calculus AB Unit 2 introduces the derivative itself, and the Unit 2 Progress Check MCQ Part A is the first real test of whether you understand what a derivative is rather than how to compute one. This walkthrough covers what the questions test, where students slip, and how to work through them. Explanations only — no AP Classroom answer keys.

Updated August 2026Written by Mahmudul HasanFree · No signup

What Unit 2 covers

Unit 2 is Differentiation: Definition and Fundamental Properties. Roughly 10–12% of the AP Calculus AB exam, and the foundation for every unit after it.

The limit definition of the derivative

f′(x) = lim_h→0 [f(x+h) − f(x)]/h, and the alternate form at a point.

Derivative as slope and as rate

The tangent line slope, and the instantaneous rate of change of a quantity.

Basic rules

Power rule, constant multiple, sum and difference.

Product and quotient rules

(fg)′ = f′g + fg′ and the quotient rule, where order matters.

Derivatives of common functions

sin x, cos x, e^x, and ln x.

Differentiability and continuity

Differentiable implies continuous — but not the reverse.

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.

Limit-definition recognition

A limit is shown and you must identify which function and which point it is the derivative of. Match the pattern to f(x+h) − f(x) over h.

Tangent line problems

Find the slope at a point, then build the line. Errors usually come from evaluating f′ at the wrong x.

Product and quotient rule execution

Straightforward but unforgiving. The quotient rule has a subtraction, so order is not optional.

Graph-based reasoning

Given a graph of f, identify where f′ is positive, negative, zero, or undefined — corners and cusps matter.

Table-based estimation

Estimate a derivative from a table of values using an average rate of change.

The Progress Check FRQ

Unit 2 free response is usually short and technical: compute a derivative, find a tangent line, or interpret f′ in context with units. Points come from correct notation, showing the rule you applied, and — on contextual parts — stating what the derivative means with correct units, not just its value.

Where students lose the most points

Reversing the quotient rule

It is (low × d-high − high × d-low) over low squared. Flipping it is the single most common Unit 2 error.

Dropping the chain rule early

If an inner function appears, you need it — even though the chain rule is formally Unit 3.

Confusing f′(a) with f(a)

One is the slope, the other is the value. Tangent-line questions need both, used in the right places.

Assuming continuity means differentiability

A sharp corner is continuous but has no derivative there.

Forgetting units in context

A derivative in an applied problem has units — meters per second, dollars per item.

How to work through this unit

If a question shows you a limit and no instructions, it is almost always asking “whose derivative is this?” Identify f and the point before doing any algebra.

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 2 Progress Check cover?

The definition of the derivative, derivative as slope and rate of change, the power, product, and quotient rules, derivatives of common functions, and the link between differentiability and continuity.

What is the hardest part of Unit 2 Progress Check MCQ Part A?

Recognising the limit definition when it appears without context, and executing the quotient rule in the right order. Both are pattern-recognition skills that improve quickly with practice.

Do I need the chain rule for Unit 2?

Formally it is Unit 3, but composite functions show up, so knowing it early helps.

Do you publish Unit 2 Progress Check answers?

No. We publish walkthroughs of the reasoning and the common mistakes — never AP Classroom answer keys.