AP Calculus AB Unit 2 Practice Test: Differentiation — Definition and Fundamental Properties
Practice AP Calculus AB Unit 2 — limit definition of derivative, product and quotient rules, trig derivatives, differentiability, and higher-order derivatives with AP-style questions.
Building the Core of Differential Calculus
Unit 2 is where calculus truly begins. You move from the conceptual framework of limits into the mechanics of the derivative — what it means, how to define it precisely, and how to compute it efficiently using foundational rules. A thorough command of Unit 2 is essential because every differentiation technique in Units 3 through 5 builds directly on what is introduced here.
Topics Covered in This Practice Test
- Average rate of change vs. instantaneous rate of change
- The limit definition of the derivative: f′(a) = lim[h→0] (f(a+h) − f(a)) / h
- Differentiability vs. continuity — when a function fails to be differentiable
- The power rule, constant rule, and constant multiple rule
- The sum and difference rules for derivatives
- The product rule and quotient rule
- Derivatives of trigonometric functions (sin, cos, tan, csc, sec, cot)
- Higher-order derivatives (second derivative and beyond)
AP Exam Skills Developed Here
MCQ questions in this unit test your ability to compute derivatives quickly and accurately using the standard rules. You will also encounter questions that ask you to interpret the derivative as a rate of change in a real-world context. FRQ questions may ask you to use the limit definition to find a derivative at a specific point — a technique that requires precision and algebraic fluency. Higher-order derivatives appear in both MCQ and FRQ contexts, often linked to concavity and motion problems in later units.
Differentiability vs. Continuity
One of the most frequently tested conceptual distinctions in Unit 2 is the relationship between differentiability and continuity. A function that is differentiable at a point must be continuous there, but continuity alone does not guarantee differentiability. Corners, cusps, vertical tangents, and jump discontinuities are the classic cases where continuity holds but differentiability fails. Expect AP questions to test this distinction in both graphical and algebraic forms.
Common Errors in Unit 2
- Forgetting to apply the quotient rule correctly — especially misplacing the subtraction in the numerator
- Dropping constant multipliers when differentiating
- Confusing the derivative of cos(x) with sin(x) rather than −sin(x)
- Applying the power rule to exponential functions — the power rule applies only when the base is a variable and the exponent is a constant