Differentiation: Definition & Rules

What this unit covers

The topics below follow the published Calculus AB course framework for Unit 2. This unit is worth 10–15% of the exam, so budget your time against that rather than against how long the unit takes to teach.

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
• Differentiability and Where Derivatives Fail14 min · 3 objectivesState the relationship between differentiability and continuity in the correct direction · Identify corners, cusps, vertical tangents, and discontinuities as failures of differentiability · Use one-sided derivatives to test differentiability of a piecewise function
• Derivatives from Graphs and Tables14 min · 3 objectivesSketch the graph of f′ from the graph of f and read the correspondence in both directions · Estimate a derivative from a table using a difference quotient · Apply the rules of differentiation to values given only in a table

Formulas in Unit 2

Every term in Unit 2

All 39 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.

Differentiability implies continuity, one way only

Differentiable at a forces continuous at a. The converse is false, and the exam tests the converse far more often.

Contrapositive as a working tool

If f is not continuous at a, then f is not differentiable at a — usually the fastest route on a graph question, since a discontinuity is easier to spot.

Corner

One-sided derivatives finite but unequal, as for |x| at 0. The function is continuous there; only the derivative fails.

Cusp

One-sided slopes go to +∞ and −∞, as for x^(2/3) at 0. Continuous, not differentiable.

Vertical tangent

Slope grows without bound with the same sign from both sides, as for x^(1/3) at 0. A tangent line exists but is vertical, so the derivative is not a number.

Testing a piecewise function at the seam

Check continuity first by matching values, then match the branch derivatives. Two conditions, which is why such problems supply two unknowns.

Smooth join is the tangent line

Joining a curve to a line differentiably makes that line the tangent at the seam. If your line differs, look for an arithmetic error.

One-sided derivative

The difference quotient limit taken from one side only. Both must exist, be finite, and agree for f′(a) to exist.

f increasing ⟺ f′ positive

The first entry of the graph-reading dictionary. Sign of f′ governs direction of f.

f concave up ⟺ f′ increasing

Concavity of f is direction of f′. This is the translation students most often invert.

Inflection point of f ⟺ extremum of f′

Where f changes concavity, f′ turns around. Useful when the graph you are given is f′.

Label which function is graphed

Write at the top of your work whether the picture is f or f′. This single line prevents the most common error in Units 2 and 5.

Maximum of f from a graph of f′

Where f′ crosses from positive to negative — NOT where the displayed curve is highest, which is where f increases fastest.

Symmetric difference quotient

f′(a) ≈ [f(a + h) − f(a − h)]/(2h). The divisor is 2h; dividing by h doubles the answer.

Estimate versus find

A table supports approximate or estimate. Claiming to have found f′(3) from a table overstates what the data allows.

Table problems test the rules

With no formula available, product and quotient rules must be written in general form first, then substituted. Substituting first is where terms get dropped.

Derivative of a constant multiple

d/dx[cf] = c·f′. Constants pass through differentiation untouched, which is why they can be pulled out first.

Derivative of a sum

Differentiation is linear: the derivative of a sum is the sum of the derivatives. No analogous rule exists for products.

Tangent line equation

y − f(a) = f′(a)(x − a). The point comes from f, the slope from f′ — mixing them up is a frequent slip.

Normal line

Perpendicular to the tangent, so its slope is −1/f′(a). Undefined when f′(a) = 0, where the normal is vertical.

Second derivative notation

f″(x), d²y/dx². Units are units of y per unit of x squared — feet per second per second for a position function of time.

Derivative of a piecewise function is piecewise

Differentiate each branch on its own open interval, then handle the breakpoint separately with one-sided derivatives.

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.
• Table problems test whether you know the **rules**, not whether you can differentiate a formula — there is no formula. Write the general rule first, then substitute. Students who substitute first almost always drop a term of the product rule or reverse the subtraction in the quotient rule.

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.

Questions about this unit

All 8 units of AP Calculus AB

1. Unit 1 · Limits & Continuity
2. Unit 2 · Differentiation: Definition & Rules
3. Unit 3 · Composite & Implicit Differentiation
4. Unit 4 · Contextual Applications
5. Unit 5 · Analytical Applications
6. Unit 6 · Integration & Accumulation
7. Unit 7 · Differential Equations
8. Unit 8 · Applications of Integration