• 53 questions

AP Calculus Unit 3 Practice Test

Unofficial practice for Unit 3 of AP Calculus AB and BC: chain rule, implicit differentiation, inverse functions and higher-order derivatives.

What is the derivative of f(x) = (3x² − 4)⁵?

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

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Unofficial practice questions, not affiliated with the College Board. Bank checked October 13, 2026.

Next step AP Calculus derivatives: Units 2 and 3AP Calculus integrals: Unit 6

All 53 questions

Topic 3.1: The chain rule 13

  • A student writes that the derivative of sin(3x²) is cos(3x²). What is wrong with the work? Wikipedia, "Chain rule"; course topic 3.1
  • Air temperature T falls 6 °C for each kilometer of altitude h, so dT/dh = −6. A hiker climbs at dh/dt = 0.5 km per hour. What is dT/dt for the hiker? Wikipedia, "Chain rule"; course topic 3.1
  • If f(x) = √(x² + 9), what is f′(4)? Wikipedia, "Chain rule"; course topic 3.1
  • If y = u³ and u = 2x + 1, what is dy/dx at x = 1? Wikipedia, "Chain rule"; course topic 3.1
  • Let k(x) = g(f(x)), where f(1) = 3, f′(1) = −2, g′(1) = 7 and g′(3) = 4. What is k′(1)? Wikipedia, "Chain rule"; course topic 3.1
  • To differentiate h(x) = cos(x⁴) as f(g(x)), which outer function f and inner function g are used? Wikipedia, "Chain rule"; course topic 3.1
  • What is the derivative of f(x) = (3x² − 4)⁵? Wikipedia, "Chain rule"; course topic 3.1
  • What is the derivative of f(x) = cos(5x)? Wikipedia, "Chain rule"; course topic 3.1
  • What is the derivative of f(x) = e^(sin x)? Wikipedia, "Chain rule"; course topic 3.1
  • What is the derivative of f(x) = e^(x²)? Wikipedia, "Chain rule"; course topic 3.1
  • What is the derivative of f(x) = ln(cos x)? Wikipedia, "Chain rule"; course topic 3.1
  • What is the derivative of f(x) = sin(e^(2x))? Wikipedia, "Chain rule"; course topic 3.1
  • What is the derivative of f(x) = tan² x? Wikipedia, "Chain rule"; course topic 3.1
Show all 53 questions (40 more)

Topic 3.2: Implicit differentiation 9

  • At which value of x does the circle x² + y² − 4x = 5 have a horizontal tangent line? Wikipedia, "Implicit function"; course topic 3.2
  • For the curve x² + xy = 10, what is dy/dx at the point (2, 3)? Wikipedia, "Implicit function"; course topic 3.2
  • For the curve x³ + y³ = 9, what is dy/dx at the point (1, 2)? Wikipedia, "Implicit function"; course topic 3.2
  • For the curve y² = x³ + 3x, what is dy/dx? Wikipedia, "Implicit function"; course topic 3.2
  • For the ellipse 4x² + y² = 8, what is dy/dx in terms of x and y? Wikipedia, "Implicit function"; course topic 3.2
  • If sin y = x, what is dy/dx in terms of y? Wikipedia, "Implicit function"; course topic 3.2
  • If y³ + y = x, what is dy/dx? Wikipedia, "Implicit function"; course topic 3.2
  • What is the slope of the line tangent to the ellipse 4x² + y² = 8 at the point (1, 2)? Wikipedia, "Implicit function"; course topic 3.2
  • When y is a function of x, what is the derivative of y² with respect to x? Wikipedia, "Implicit function"; course topic 3.2

Topic 3.3: Differentiating inverse functions 8

  • A differentiable function f has an inverse g, with f(2) = 6, f′(2) = −4, f(6) = 1 and f′(6) = −3. What is g′(1)? Wikipedia, "Inverse function rule"; course topic 3.3
  • If g is the inverse of a differentiable function f, which expression gives g′(x) wherever it exists? Wikipedia, "Inverse function rule"; course topic 3.3
  • If g is the inverse of f, then f(g(x)) = x. What does differentiating both sides with the chain rule give? Wikipedia, "Inverse function rule"; course topic 3.3
  • Let f(x) = 2x + sin x and let g be the inverse of f. Since f(0) = 0, what is g′(0)? Wikipedia, "Inverse function rule"; course topic 3.3
  • Let f(x) = e^x + x and let g be the inverse of f. Since f(0) = 1, what is g′(1)? Wikipedia, "Inverse function rule"; course topic 3.3
  • Let f(x) = x³, with inverse g(x) = ∛x. What is g′(8)? Wikipedia, "Inverse function rule"; course topic 3.3
  • Let f(x) = x⁵ + 3x + 2 and let g be the inverse of f. Since f(0) = 2, what is g′(2)? Wikipedia, "Inverse function rule"; course topic 3.3
  • The line tangent to the graph of f at the point (3, 7) has slope 4. What is the slope of the line tangent to the graph of f⁻¹ at (7, 3)? Wikipedia, "Inverse function rule"; course topic 3.3

Topic 3.4: Differentiating inverse trigonometric functions 8

  • If f(x) = arccos(2x), what is f′(0)? Wikipedia, "Inverse trigonometric functions"; course topic 3.4
  • If f(x) = arctan x, what is f′(1)? Wikipedia, "Inverse trigonometric functions"; course topic 3.4
  • If y = arcsin x, implicit differentiation of sin y = x gives dy/dx = 1/cos y. What is cos y in terms of x? Wikipedia, "Inverse trigonometric functions"; course topic 3.4
  • What is the derivative of f(x) = arccos x? Wikipedia, "Inverse trigonometric functions"; course topic 3.4
  • What is the derivative of f(x) = arcsin(3x)? Wikipedia, "Inverse trigonometric functions"; course topic 3.4
  • What is the derivative of f(x) = arcsin(x/2)? Wikipedia, "Inverse trigonometric functions"; course topic 3.4
  • What is the derivative of f(x) = arctan(e^x)? Wikipedia, "Inverse trigonometric functions"; course topic 3.4
  • What is the derivative of f(x) = arctan(x²)? Wikipedia, "Inverse trigonometric functions"; course topic 3.4

Topic 3.5: Selecting procedures for calculating derivatives 7

  • If f(x) = x√(2x + 1), what is f′(4)? Wikipedia, "Differentiation rules"; course topic 3.5
  • Let h(x) = f(x) · g(2x), where f(1) = 2, f′(1) = 3, g(2) = 5 and g′(2) = −1. What is h′(1)? Wikipedia, "Differentiation rules"; course topic 3.5
  • To differentiate y = ln(x sin x), which rule is applied first? Wikipedia, "Differentiation rules"; course topic 3.5
  • What is the derivative of f(x) = (ln x)²? Wikipedia, "Differentiation rules"; course topic 3.5
  • What is the derivative of f(x) = e^(2x) cos x? Wikipedia, "Differentiation rules"; course topic 3.5
  • What is the derivative of f(x) = ln(x³) for x > 0? Wikipedia, "Differentiation rules"; course topic 3.5
  • What is the derivative of f(x) = x arctan x? Wikipedia, "Differentiation rules"; course topic 3.5

Topic 3.6: Calculating higher-order derivatives 8

  • For the circle x² + y² = 4, implicit differentiation gives dy/dx = −x/y. What is d²y/dx² in terms of y? Wikipedia, "Second derivative"; course topic 3.6
  • For y = f(x), which notation stands for the second derivative of y with respect to x? Wikipedia, "Second derivative"; course topic 3.6
  • What is the fourth derivative of f(x) = cos x? Wikipedia, "Derivative"; course topic 3.6
  • What is the second derivative of f(x) = 1/x? Wikipedia, "Second derivative"; course topic 3.6
  • What is the second derivative of f(x) = xe^x? Wikipedia, "Second derivative"; course topic 3.6
  • What is the second derivative of f(x) = x⁴ − 3x² + x? Wikipedia, "Second derivative"; course topic 3.6
  • What is the second derivative of y = e^(3x)? Wikipedia, "Second derivative"; course topic 3.6
  • What is the third derivative of f(x) = x⁵? Wikipedia, "Derivative"; course topic 3.6

How this list was compiled: Our own questions, written to Unit 3 (Differentiation: Composite, Implicit, and Inverse Functions; topics 3.1 to 3.6) 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 2026-10-13. Every answer is mathematics worked from what the question prints, and the computed ones were worked out a second time with computer algebra on 2026-10-13. The rule behind each answer was read on 2026-10-13 in the Wikipedia article named under the question. Licence: Own work.

How it works

This AP Calculus Unit 3 practice test has 53 questions of our own on Differentiation: Composite, Implicit, and Inverse Functions. It is unofficial and is not affiliated with or endorsed by the College Board, which owns AP.

Read the full guide

What Unit 3 covers

The College Board's course description for AP Calculus AB and BC, read October 13, 2026, gives Unit 3 six topics, and each has its own set here: the chain rule (13 questions), implicit differentiation (9), inverse functions (8), inverse trigonometric functions (8), selecting procedures (7) and higher-order derivatives (8). The description gives topics no weights, so the spread is ours.

Worked answers

Every answer is worked from the function in the question, and the computed ones were worked out a second time with computer algebra. Where one wrong choice is the usual slip, the explanation says which.

  • Say "times the derivative of the inside" each time you use the chain rule. It is the step most often left out.
  • In implicit differentiation every y term picks up a factor of dy/dx. Collect those terms on one side before you solve.
  • For an inverse, find the point first: g′(a) is 1 over f′ at the x where f(x) = a.

Questions and answers

What is Unit 3 in AP Calculus?

Differentiation: Composite, Implicit, and Inverse Functions. It has six topics, from the chain rule (3.1) to higher-order derivatives (3.6).

How much of the AP Calculus exam is Unit 3?

From 5 to 10% of the multiple-choice section on both the AB and the BC exam, by the College Board's course pages. The chain rule also turns up inside questions from later units.

Is Unit 3 the same in AP Calculus AB and BC?

Yes. The unit and its six topics belong to both courses. The course description suggests 10 to 11 class periods for it in AB and 8 to 9 in BC.

Is this the AP Calculus Unit 3 progress check?

No. The College Board's Progress Check for Unit 3, about 15 multiple-choice and 3 free-response questions by the course description, is assigned by teachers in AP Classroom. These questions are our own.

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