Differentiation: Definition & Fundamental Properties

Name: AP Calculus BC Unit 2: Differentiation: Definition & Fundamental Properties
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4–7% of the exam1 lesson · 14 min4 terms

What this unit covers

The topics below follow the published Calculus BC course framework for Unit 2. This unit is worth 4–7% of the exam, so budget your time against that rather than against how long the unit takes to teach.

Definition of the derivativePower ruleProduct & quotientTrig derivatives

Lessons in this unit

The Derivative & Basic Rules14 min · 3 objectivesDefine the derivative as a limit of a difference quotient · Apply the power, constant-multiple, and sum rules · Differentiate products and quotients with the product and quotient rules

Formulas in Unit 2

Limit definition of the derivative

f′(x) = lim(h→0) [f(x + h) − f(x)] / h

The difference quotient [f(x+h) − f(x)]/h is the secant slope; its limit as h→0 is the tangent slope.

Product and quotient rules

(fg)′ = f′g + fg′ · (f/g)′ = (f′g − fg′) / g²

The quotient rule numerator is "low d-high minus high d-low"; the order of subtraction matters.

Every term in Unit 2

All 4 terms we publish for Differentiation: Definition & Fundamental Properties, with definitions. Reading them through is the fastest way to find the ones you cannot define — then drill those in cram mode until you can produce them without the prompt.

Derivatives of inverse trig functions: arcsin′ = 1/√(1 − x²), arctan′ = 1/(1 + x²), arcsec′ = 1/(|x|√(x² − 1)).
Logarithmic differentiation: Take ln of both sides before differentiating. Required for a variable base raised to a variable exponent.
Derivative of an inverse function: (f⁻¹)′(b) = 1/f′(a) where f(a) = b. Locate the input a first; the formula uses f′ there, not at b.
Higher-order derivatives in motion: Position, velocity, acceleration and jerk are successive derivatives. Speed is the magnitude of velocity, not its derivative.

What examiners penalize here

When a limit is written as lim(h→0) [f(a+h) − f(a)]/h, recognize it as the *definition of a derivative at a point*, f′(a). AP loves disguising a derivative as a raw limit — identifying it lets you skip the algebra and just differentiate.

Practice Calculus BC

Our practice bank is drawn from across the whole course rather than filtered to one unit, which is closer to how the exam asks anyway — it will not tell you which unit a question is testing.

42 practice questionsEvery one with a written explanation FRQ trainerWritten answers scored against the rubric Calculus BC study guideAll units, weights and score data in one place All 10 Calculus BC unitsThe full course

Questions about this unit

How much of the AP Calculus BC exam is Unit 2?

Unit 2, Differentiation: Definition & Fundamental Properties, is worth 4–7% of the Calculus BC multiple-choice section according to the published course framework. Across all 10 units that makes it one of the lighter units, so it is not where a review phase should start.

What topics are covered in Calculus BC Unit 2?

Differentiation: Definition & Fundamental Properties covers Definition of the derivative, Power rule, Product & quotient and Trig derivatives. We publish 4 terms with definitions for this unit, all of them on this page.

How should I study Calculus BC Unit 2?

Read the 1 lesson below first — about 15 minutes — then drill the 4 terms in cram mode until you can produce each definition from memory rather than just recognize it. Recognition is what makes a unit feel finished when it is not. Finish with practice questions and read the explanation for every one you get right by elimination as well as the ones you miss.

All 10 units of AP Calculus BC

Unit names, topics and exam weights follow the published College Board course framework for AP Calculus BC. AP® is a trademark registered by the College Board, which does not endorse this site.