• 51 questions

AP Calculus AB Derivatives Practice Test

Unofficial AP Calculus AB derivative practice: the definition, the rules, the chain rule, implicit differentiation and the derivative tests.

What is the derivative of f(x) = 4x⁵?

Pick an answer, then press Check answer. Then you see Correct or Not quite, with the reason. Keys: A to D pick, Enter checks.

Print covers the next 25 questions. Share this set copies a link to the same questions in the same order.

More options

Unofficial practice questions, not affiliated with the College Board. Bank checked October 10, 2026.

Next step Calculus questions on every topicAP Physics C rotation practice

All 51 questions

Topics 2.1 to 2.4: Defining the derivative 9

  • A differentiable function has f(1.9) = 4.61 and f(2.1) = 5.41. What is the best estimate of f′(2) from these values? Wikipedia, "Derivative"; course topic 2.3
  • At which value of x is f(x) = |x − 2| continuous but not differentiable? Wikipedia, "Differentiable function"; course topic 2.4
  • Let f(x) = x² for x ≤ 1 and f(x) = 2x − 1 for x > 1. Is f differentiable at x = 1? Wikipedia, "Differentiable function"; course topic 2.4
  • What is the average rate of change of f(x) = x² + 3x on the interval from x = 1 to x = 4? Wikipedia, "Derivative"; course topic 2.1
  • What is the slope of the line tangent to the graph of f(x) = x³ − 4x at x = 2? Wikipedia, "Derivative"; course topic 2.2
  • What is the value of the limit of (√(16 + h) − 4)/h as h approaches 0? Wikipedia, "Derivative" and Wikipedia, "Power rule"; course topic 2.2
  • Which expression defines the derivative of f at x = a? Wikipedia, "Derivative"; course topic 2.2
  • Which is an equation of the line tangent to y = x³ at the point where x = 2? Wikipedia, "Derivative"; course topic 2.2
  • Which statement about a function f and a point x = a is always true? Wikipedia, "Differentiable function"; course topic 2.4
Show all 51 questions (42 more)

Topics 2.5 to 2.10: Derivative rules 15

  • At which values of x does the graph of f(x) = x³ − 12x have a horizontal tangent line? Wikipedia, "Derivative" and Wikipedia, "Power rule"; course topic 2.6
  • Functions f and g are differentiable with f(2) = 3, f′(2) = −1, g(2) = 4 and g′(2) = 5. If h(x) = f(x)g(x), what is h′(2)? Wikipedia, "Product rule"; course topic 2.8
  • What is the derivative of f(x) = (2x + 1)/(x − 3)? Wikipedia, "Quotient rule"; course topic 2.9
  • What is the derivative of f(x) = 3 sin x − 2 cos x? Wikipedia, "Differentiation of trigonometric functions"; course topic 2.7
  • What is the derivative of f(x) = 3/x²? Wikipedia, "Power rule"; course topic 2.5
  • What is the derivative of f(x) = 4x⁵? Wikipedia, "Power rule"; course topic 2.5
  • What is the derivative of f(x) = 5eˣ + 2 ln x? Wikipedia, "Differentiation rules"; course topic 2.7
  • What is the derivative of f(x) = sec x? Wikipedia, "Differentiation of trigonometric functions"; course topic 2.10
  • What is the derivative of f(x) = tan x? Wikipedia, "Differentiation of trigonometric functions"; course topic 2.10
  • What is the derivative of f(x) = x/(x² + 1)? Wikipedia, "Quotient rule"; course topic 2.9
  • What is the derivative of f(x) = x² ln x? Wikipedia, "Product rule" and Wikipedia, "Differentiation rules"; course topic 2.8
  • What is the derivative of f(x) = x³ cos x? Wikipedia, "Product rule"; course topic 2.8
  • What is the derivative of f(x) = x³ − 6x² + 2x − 7? Wikipedia, "Linearity of differentiation" and Wikipedia, "Power rule"; course topic 2.6
  • What is the derivative of f(x) = √x? Wikipedia, "Power rule"; course topic 2.5
  • With f(2) = 3, f′(2) = −1, g(2) = 4 and g′(2) = 5, let k(x) = f(x)/g(x). What is k′(2)? Wikipedia, "Quotient rule"; course topic 2.9

Topics 3.1 and 3.2: Chain rule and implicit differentiation 11

  • For the circle x² + y² = 169, what is dy/dx in terms of x and y? Wikipedia, "Implicit function"; course topic 3.2
  • For the curve xy + y² = 6, what is dy/dx at the point (1, 2)? Wikipedia, "Implicit function" and Wikipedia, "Product rule"; course topic 3.2
  • If f(x) = √(4x + 9), what is f′(4)? Wikipedia, "Chain rule"; course topic 3.1
  • Let h(x) = f(g(x)), where g(1) = 3, g′(1) = 2, f′(1) = 4 and f′(3) = 5. What is h′(1)? Wikipedia, "Chain rule"; course topic 3.1
  • What is the derivative of f(x) = (2x − 5)⁴? Wikipedia, "Chain rule"; course topic 3.1
  • What is the derivative of f(x) = cos³ x? Wikipedia, "Chain rule" and Wikipedia, "Differentiation of trigonometric functions"; course topic 3.1
  • What is the derivative of f(x) = e^(−2x)? Wikipedia, "Chain rule" and Wikipedia, "Differentiation rules"; course topic 3.1
  • What is the derivative of f(x) = ln(3x² + 2)? Wikipedia, "Chain rule" and Wikipedia, "Differentiation rules"; course topic 3.1
  • What is the derivative of f(x) = sin(x³)? Wikipedia, "Chain rule"; course topic 3.1
  • What is the derivative of f(x) = xe^(3x)? Wikipedia, "Product rule" and Wikipedia, "Chain rule"; course topic 3.1
  • What is the slope of the line tangent to the circle x² + y² = 169 at the point (5, 12)? Wikipedia, "Implicit function"; course topic 3.2

Topics 3.3 to 3.6: Inverse functions and higher derivatives 8

  • A differentiable function f has an inverse g. If f(3) = 5, f′(3) = 2 and f′(5) = 7, what is g′(5)? Wikipedia, "Inverse function rule"; course topic 3.3
  • Let f(x) = x³ + x and let g be the inverse of f. Since f(1) = 2, what is g′(2)? Wikipedia, "Inverse function rule"; course topic 3.3
  • What is the derivative of f(x) = arcsin x? Wikipedia, "Differentiation of trigonometric functions"; course topic 3.4
  • What is the derivative of f(x) = arctan(2x)? Wikipedia, "Differentiation of trigonometric functions" and Wikipedia, "Chain rule"; course topic 3.4
  • What is the second derivative of f(x) = x⁵ − 4x³? Wikipedia, "Second derivative"; course topic 3.6
  • What is the second derivative of y = ln x? Wikipedia, "Second derivative" and Wikipedia, "Differentiation rules"; course topic 3.6
  • What is the second derivative of y = sin(2x)? Wikipedia, "Second derivative" and Wikipedia, "Chain rule"; course topic 3.6
  • Which rules are needed to differentiate y = x² sin(3x)? Wikipedia, "Product rule" and Wikipedia, "Chain rule"; course topic 3.5

Topics 5.4 and 5.7: First and second derivative tests 8

  • A function f has derivative f′(x) = (x − 1)(x − 4). At which value of x does f have a local maximum? Wikipedia, "Derivative test"; course topic 5.4
  • A function f has derivative f′(x) = x²(x − 2). What does the first derivative test say about f at x = 0? Wikipedia, "Derivative test"; course topic 5.4
  • At which value of x does f(x) = xe^(−x) have a local maximum? Wikipedia, "Derivative test" and Wikipedia, "Product rule"; course topic 5.4
  • At which value of x does f(x) = x³ − 3x² − 9x + 5 have a local minimum? Wikipedia, "Derivative test"; course topic 5.4
  • For f(x) = x⁴, both f′(0) and f″(0) equal 0. What follows? Wikipedia, "Derivative test"; course topic 5.7
  • For x > 0, the function f(x) = x + 4/x has one critical point. What is the local minimum value of f? Wikipedia, "Derivative test"; course topic 5.7
  • The function f(x) = 2x³ − 9x² + 12x has a critical point at x = 1. What does the second derivative test show there? Wikipedia, "Derivative test"; course topic 5.7
  • Suppose f′(c) = 0 and f″(c) > 0. What does the second derivative test conclude? Wikipedia, "Derivative test"; course topic 5.7

How this list was compiled: Our own questions, written to Unit 2 (Differentiation: Definition and Fundamental Properties), Unit 3 (Differentiation: Composite, Implicit, and Inverse Functions) and topics 5.4 and 5.7 (the first and second derivative tests) of the College Board's AP Calculus AB and BC Course and Exam Description, https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf, read 2026-10-10. Every answer is mathematics worked from what the question prints, and the computed ones were worked out a second time with computer algebra on 2026-10-10. The rule behind each answer was read on 2026-10-10 in the Wikipedia article named under the question. Licence: Own work.

How it works

This AP Calculus AB derivatives practice test has 51 questions of our own on Units 2 and 3, plus the first and second derivative tests. It is unofficial and not affiliated with or endorsed by the College Board.

Read the full guide

What the questions cover

The bank follows the College Board's course description, read October 10, 2026. Unit 2, Differentiation: Definition and Fundamental Properties, has 24 questions: 15 on the rules and 9 on the definition. Unit 3, Composite, Implicit, and Inverse Functions, has 19. Eight more cover topics 5.4 and 5.7 of Unit 5, the derivative tests. The description gives topics no weights, so the spread is ours.

Worked answers

Each answer is worked from the function in the question, and the computed ones were worked out a second time with computer algebra. The explanation shows the steps and names the slip behind a tempting wrong choice.

  • Before you differentiate, name the outermost operation: a product, a quotient or a composition. That picks the rule.
  • In a table problem on f(g(x)), take the outer derivative at g(x), not at x.
  • If f″ is zero at a critical point, the second derivative test says nothing. Go back to the sign of f′.

Questions and answers

What units are derivatives in AP Calculus AB?

Units 2 and 3 teach how to differentiate. By the College Board's course page, Unit 2 is 10 to 15% of the multiple-choice section and Unit 3 is 5 to 10%. Units 4 and 5 then apply derivatives.

Does this work as an AP Calc BC derivative practice test?

Yes. Units 2 and 3 and topics 5.4 and 5.7 belong to both AB and BC. BC adds derivatives of parametric, polar and vector-valued functions in Unit 9, which this page does not cover.

Where can I practice the first derivative and second derivative test for AP Calculus AB?

Here, in the last topic set. Four questions use the sign of f′ to find a maximum or a minimum, and four use f″, including a case where the second derivative test is inconclusive.

Can you use a calculator on AP Calculus derivative questions?

None is needed here. On the exam, Part A of the multiple-choice section is 29 questions in 62 minutes with no calculator, and Part B is 13 questions in 38 minutes with a graphing calculator.

Are these real AP Calculus exam questions?

No. The College Board owns the real questions, and none appears here, even in other words. Every question was written for this page from the topics the course description lists.

More tests

Show 8 more

Browse Civics and Law