SAT / PSAT
SAT / PSAT Prep
History & Social Science
AP World History AP US History AP European History AP Human Geography AP US Government & Politics AP Psychology AP Macroeconomics AP Microeconomics
English
AP English Language & Composition AP English Literature & Composition
Math & Computer Science
AP Calculus AB/BC AP Precalculus AP Statistics AP Computer Science A AP Computer Science Principles
Sciences
AP Biology AP Chemistry AP Environmental Science AP Physics 1 AP Physics 2
World Languages & Arts
AP Spanish Language AP Art History AP Music Theory Start studying →
Unit 2 · Definition of the Derivative Flashcards Cheat Sheet Essentials Visual Review MC Practice FRQ Practice

AP Calculus AB/BC Unit 2 Essentials

The must-know terms and core concepts for Unit 2: Definition of the Derivative. Every vocabulary word, rule, and idea you need to master.

← Back to Unit 2 hub
Key Concept 1
The derivative is an instantaneous rate of change — a limit of average rates
The average rate of change of f on an interval is the slope of the secant line, [f(b) − f(a)]/(b − a). As the interval shrinks to a single point, that average approaches the instantaneous rate of change — the derivative, f′(x) = lim(h→0) [f(x+h) − f(x)]/h. Geometrically, f′(a) is the slope of the tangent line at (a, f(a)), which lets you write the tangent line and estimate the derivative from graphs and tables.
Rates of Change Limit Definition Tangent Slope
Key Concept 2
Differentiability requires continuity and smoothness
If a function is differentiable at a point, it must be continuous there — but the converse fails. A derivative does not exist where a graph has a corner (like |x| at 0), a cusp, a vertical tangent, or a discontinuity. Recognizing these cases lets you decide where a function can be differentiated and interpret what a graph's behavior means for its rate of change.
Differentiability Continuity Where Derivatives Fail
Key Concept 3
The fundamental rules make differentiation fast
Rather than compute a limit every time, you use rules. The power rule d/dx[xⁿ] = n·xⁿ⁻¹, along with the constant, constant-multiple, sum, and difference rules, handles polynomials. Memorize the derivatives of sin x, cos x, eᵇ, and ln x, then combine functions with the product rule (fg)′ = f′g + fg′ and the quotient rule (f/g)′ = (f′g − fg′)/g² — which also give the derivatives of the remaining trig functions.
Power Rule Key Derivatives Product & Quotient
Average rate of change
The slope of the secant line over [a, b]: [f(b) − f(a)]/(b − a).
Rates
Instantaneous rate of change
The derivative at a point; the limit of average rates as the interval shrinks.
Rates
Difference quotient
The expression [f(x + h) − f(x)]/h whose limit defines the derivative.
The Derivative
Derivative
f′(x) = lim(h→0) [f(x+h) − f(x)]/h, the instantaneous rate of change.
The Derivative
Derivative notation
f′(x), y′, dy/dx, or d/dx[f(x)].
The Derivative
Tangent line
The line through (a, f(a)) with slope f′(a): y − f(a) = f′(a)(x − a).
The Derivative
Differentiability
Having a derivative at a point; requires a smooth, non-vertical tangent.
Differentiability
Differentiability implies continuity
A function differentiable at c is continuous at c (but not conversely).
Differentiability
Corner / cusp
Points where left and right slopes differ, so the derivative does not exist.
Differentiability
Vertical tangent
A point where the tangent is vertical and the derivative is undefined.
Differentiability
Power Rule
d/dx[xⁿ] = n·xⁿ⁻¹.
Rules
Constant Rule
d/dx[c] = 0.
Rules
Constant Multiple Rule
d/dx[c·f(x)] = c·f′(x).
Rules
Sum / Difference Rule
d/dx[f ± g] = f′ ± g′.
Rules
Derivative of sin x / cos x
cos x and −sin x, respectively.
Key Derivatives
Derivative of eᵇ / ln x
eᵇ and 1/x, respectively.
Key Derivatives
Product Rule
d/dx[f·g] = f′g + fg′.
Product & Quotient
Quotient Rule
d/dx[f/g] = (f′g − fg′)/g².
Product & Quotient
Derivative of tan x
sec²x.
Trig Derivatives
Derivatives of sec, cot, csc
sec x tan x, −csc²x, and −csc x cot x.
Trig Derivatives