Differentiation: Definition & Rules

Name: AP Calculus AB Unit 2: Differentiation: Definition & Rules
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10–12% of the exam4 lessons · 56 min17 terms

What this unit covers

The topics below follow the published Calculus AB course framework for Unit 2. This unit is worth 10–12% of the exam, so budget your time against that rather than against how long the unit takes to teach.

Power ruleProduct & quotientChain ruleTrig derivatives

Lessons in this unit

The Derivative as a Limit14 min · 3 objectivesInterpret the derivative as the slope of the tangent line and an instantaneous rate of change · Compute a derivative from the limit definition (difference quotient) · Relate differentiability to continuity
The Power Rule & Basic Rules13 min · 3 objectivesDifferentiate power functions with the power rule, including negative and fractional exponents · Apply the constant-multiple and sum/difference rules · Rewrite roots and reciprocals as powers before differentiating
The Product & Quotient Rules14 min · 3 objectivesDifferentiate products of functions with the product rule · Differentiate quotients with the quotient rule · Choose between rewriting and applying a rule based on the expression
The Chain Rule & Trig Derivatives15 min · 3 objectivesDifferentiate composite functions with the chain rule · State and apply the derivatives of the six trigonometric functions · Combine the chain rule with trig, power, and product rules

Formulas in Unit 2

Definition of the derivative

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

Equivalent form at a point: f′(a) = lim(x→a) [f(x) − f(a)]/(x − a). Both compute the tangent slope.

Power rule

d/dx[xⁿ] = n·xⁿ⁻¹

Works for every constant n. Special cases: d/dx[c] = 0 and d/dx[x] = 1.

Linearity of the derivative

d/dx[c·f(x)] = c·f′(x) and d/dx[f(x) ± g(x)] = f′(x) ± g′(x)

Constants pass through; sums and differences split apart. Differentiate a polynomial term by term.

Product rule

d/dx[f·g] = f′·g + f·g′

Derivative of first times second, plus first times derivative of second. Not f′·g′.

Quotient rule

d/dx[f/g] = (f′·g − f·g′) / g²

Low d-high minus high d-low, all over low squared. The order of subtraction is essential.

Chain rule

d/dx[f(g(x))] = f′(g(x)) · g′(x)

Outer derivative (inside untouched) times inner derivative. In Leibniz form, dy/dx = dy/du · du/dx.

Core trig derivatives

d/dx[sin x] = cos x · d/dx[cos x] = −sin x · d/dx[tan x] = sec²x

Also: d/dx[sec x] = sec x·tan x, d/dx[csc x] = −csc x·cot x, d/dx[cot x] = −csc²x.

Every term in Unit 2

All 17 terms we publish for Differentiation: Definition & Rules, 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.

Product rule: d/dx[uv] = u′v + uv′. The derivative of a product is never the product of the derivatives.
Quotient rule: d/dx[u/v] = (u′v − uv′)/v². Order matters in the numerator; reversing it flips the sign.
Derivative as a limit: f′(x) = lim(h→0) [f(x + h) − f(x)]/h, the limit of the average rate of change as the interval shrinks.
Alternate form of the definition: f′(a) = lim(x→a) [f(x) − f(a)]/(x − a). Recognizing a given limit as this form identifies the function and the point.
Derivative as slope of the tangent: f′(a) is the slope of the tangent line at x = a; the tangent line is y = f(a) + f′(a)(x − a).
Differentiability implies continuity: The converse is false — |x| is continuous at 0 but not differentiable there. Corners, cusps, vertical tangents and discontinuities all block differentiability.
Power rule: d/dx[xⁿ] = nxⁿ⁻¹, valid for every real n including negative and fractional exponents.
Derivatives of trig functions: sin→cos, cos→−sin, tan→sec², sec→sec·tan, csc→−csc·cot, cot→−csc². The co-functions all carry a minus sign.
Derivative of eˣ and ln x: d/dx[eˣ] = eˣ and d/dx[ln x] = 1/x for x > 0.
Derivative of aˣ and log_a x: d/dx[aˣ] = aˣ ln a and d/dx[log_a x] = 1/(x ln a).
Higher-order derivatives: f″ is the derivative of f′ and gives concavity; in motion, position, velocity and acceleration are successive derivatives.
Recognizing a limit as a derivative: A limit matching [f(a + h) − f(a)]/h is f′(a). Free-response questions ask this without saying the word "derivative".
Where a function fails to be differentiable: Corners, cusps, vertical tangents and discontinuities. Sketch or check one-sided derivatives rather than assuming.
Tangent vs normal line: The tangent has slope f′(a); the normal is perpendicular, with slope −1/f′(a).
Using a tangent line to approximate: Substitute the nearby x into y = f(a) + f′(a)(x − a). State whether it over- or underestimates using concavity.
Derivative from a table of values: Approximate with a difference quotient between the two closest values, and say the approximation is an average rate of change.
Product rule with three factors: Differentiate each factor in turn while holding the others fixed, then add the three terms.

What examiners penalize here

When a problem says "using the definition of the derivative", you must show the limit of the difference quotient — the shortcut rules earn no credit on that specific prompt. Look for the h → 0 phrasing as your cue.
Watch the sign and the new exponent carefully with negative powers. d/dx[x⁻²] = −2x⁻³ — the coefficient is negative *and* the exponent becomes more negative. Rushing this step is a frequent multiple-choice trap.
Before reaching for the quotient rule, ask whether the denominator is a single term you can divide out. Rewriting (x³ + x)/x as x² + 1 turns a quotient-rule problem into a one-line power-rule problem and removes a chance to slip on signs.
Think of the chain rule as peeling an onion: differentiate the outermost layer first, keep the inside intact, then multiply by the derivative of the inside, repeating for nested compositions. Track every layer and you will never drop a factor.

Practice Calculus AB

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.

40 practice questionsEvery one with a written explanation FRQ trainerWritten answers scored against the rubric Calculus AB study guideAll units, weights and score data in one place All 8 Calculus AB unitsThe full course

Questions about this unit

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

Unit 2, Differentiation: Definition & Rules, is worth 10–12% of the Calculus AB multiple-choice section according to the published course framework. Across all 8 units that makes it a substantial share — heavier than an even split would give it.

What topics are covered in Calculus AB Unit 2?

Differentiation: Definition & Rules covers Power rule, Product & quotient, Chain rule and Trig derivatives. We publish 17 terms with definitions for this unit, all of them on this page.

How should I study Calculus AB Unit 2?

Read the 4 lessons below first — about 55 minutes — then drill the 17 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 8 units of AP Calculus AB

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