AP Calculus AB, AP exams, 10–12%

Unit 2: Differentiation: Definition and Fundamental Properties

Unit 2 defines the derivative as a limit of difference quotients, then builds the basic rules that make computation practical. The connection between differentiability and continuity, and the standard derivatives of power, exponential, logarithmic, and trigonometric functions, are established here.

Position: 2 of 8 units
Published weighting: 10–12% in the official framework
Topics covered: 6 topics in this guide

What is covered

Topics inside this unit.

These topics describe the scope students most often review for this part of AP Calculus AB. Treat them as a study checklist rather than an exhaustive list of what a single exam form can sample.

Defining the derivative: Average rate of change, the difference quotient, and the derivative as the limit of that quotient.
Derivative as a function: Estimating derivative values from graphs and tables, and sketching a derivative from a function graph.
Differentiability and continuity: Why differentiability implies continuity, and the corners, cusps, and vertical tangents where it fails.
Basic derivative rules: Power, constant multiple, sum and difference rules applied to polynomial and radical functions.
Derivatives of transcendental functions: Derivatives of sine, cosine, the exponential function, and the natural logarithm.
Product and quotient rules: Applying both rules cleanly, including recognizing when algebraic simplification first is faster.

Study tips

How to practice Differentiation Basics.

Section-specific guidance that follows from how this material is usually tested.

Compute a derivative from the limit definition a few times each week; the exam asks for it directly.
Practice sketching the derivative graph from a function graph and vice versa until the shape relationships are automatic.
Keep a short list of the functions where differentiability fails and be able to name the reason.

Where this fits

Unit 2: Differentiation: Definition and Fundamental Properties is 2 of 8 units in the AP Calculus AB guide, and it carries a published weighting of 10–12% in the official framework. Use the full guide to balance it against everything else you have to review.

Back to the AP Calculus AB guide Official exam information

Sample Differentiation Basics questions

Practice questions, not yet expert-reviewed

Original practice questions written to the published framework for this section. No expert has reviewed them, and they are not taken from any official exam. Try each one before you open the answer.

Question 1Foundational
If f(x) = x^4 - 3x^2 + 7, what is f'(x)?
1. A4x^3 - 6x
2. B4x^3 - 6x + 7
3. C4x^3 - 3x
4. Dx^5/5 - x^3 + 7x
Show answer
Answer: A. 4x^3 - 6x
Apply the power rule term by term: the derivative of x^4 is 4x^3, the derivative of -3x^2 is -6x, and the derivative of the constant 7 is 0. Keeping the +7 treats a constant as if it had a nonzero derivative, and the last choice is an antiderivative rather than a derivative.
Key insight: The derivative of any constant term is zero, no matter how large the constant.
Question 2Foundational
What is the value of lim h→0 of ((5 + h)^2 - 25)/h?
1. A0
2. B5
3. C10
4. D25
Show answer
Answer: C. 10
Expanding gives (25 + 10h + h^2 - 25)/h = (10h + h^2)/h = 10 + h, which approaches 10. This difference quotient is the derivative of f(x) = x^2 at x = 5, and f'(x) = 2x confirms f'(5) = 10. Answering 25 reports f(5) instead of f'(5).
Key insight: A limit of the form (f(a + h) - f(a))/h is the derivative at x = a.
Question 3Intermediate
If f(x) = (2x + 1)/(x - 3), what is f'(x)?
1. A-7/(x - 3)^2
2. B7/(x - 3)^2
3. C2/(x - 3)^2
4. D(4x - 5)/(x - 3)^2
Show answer
Answer: A. -7/(x - 3)^2
The quotient rule gives [2(x - 3) - (2x + 1)(1)]/(x - 3)^2, and the numerator simplifies to 2x - 6 - 2x - 1 = -7. The answer 7/(x - 3)^2 comes from reversing the order of the two products, and (4x - 5)/(x - 3)^2 comes from adding instead of subtracting them.
Key insight: In the quotient rule the low-times-derivative-of-high term comes first, and order matters.
Question 4Intermediate
Is the following statement true or false: if a function f is continuous at x = a, then f must be differentiable at x = a?
1. ATrue
2. BFalse
Show answer
Answer: B. False
The implication runs the other way: differentiability guarantees continuity, but not the reverse. The function f(x) = |x| is continuous at x = 0 yet has no derivative there because the left and right difference quotients approach -1 and 1. Corners, cusps, and vertical tangents are the standard counterexamples.
Key insight: Differentiable implies continuous; continuous does not imply differentiable.
Question 5Advanced
If f(x) = x^3 ln(x) for x > 0, what is f'(x)?
1. A3x^2 ln(x) + x^2
2. B3x^2 ln(x) + x^3
3. C3x^2 ln(x)
4. D3x^2/x
Show answer
Answer: A. 3x^2 ln(x) + x^2
The product rule gives f'(x) = 3x^2 ln(x) + x^3 times the derivative of ln(x), which is 1/x, so the second term is x^3/x = x^2. Choosing 3x^2 ln(x) + x^3 forgets to multiply by 1/x, and choosing 3x^2 ln(x) drops the second product-rule term entirely.
Key insight: After the product rule, simplify x^3 times 1/x to x^2 rather than leaving it unreduced.

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Unit 2: Differentiation: Definition and Fundamental Properties