AP Calculus Formula Sheet: Every Derivative and Integral to Know
A complete AP Calculus formula sheet for AB and BC: limits, derivatives, integrals, key theorems, area and volume, and BC series. No sheet is given on exam day.
College Board does not provide a formula sheet on the AP Calculus AB or BC exam, so you need to memorize every derivative rule, integral rule, and theorem you will use. This page collects all of them in one place, organized the way the course is taught, with the BC-only formulas in their own section at the end. Use it as a checklist: if you cannot reproduce a row from memory, that is your next thing to drill.
Limits
| Formula | Notes |
|---|---|
| lim (x→0) sin(x)/x = 1 | Foundation for the derivative of sin x |
| lim (x→0) (1 − cos x)/x = 0 | Often appears with trig limits |
| L'Hospital's Rule: if lim f/g gives 0/0 or ∞/∞, then lim f/g = lim f'/g' | Confirm the indeterminate form first, and say so in an FRQ |
| f is continuous at a if lim (x→a) f(x) = f(a) | The limit must exist and equal the function value |
Derivative rules
| Rule | Formula |
|---|---|
| Definition | f'(x) = lim (h→0) [f(x + h) − f(x)] / h |
| Power | d/dx [xⁿ] = n·xⁿ⁻¹ |
| Product | d/dx [f·g] = f'g + fg' |
| Quotient | d/dx [f/g] = (f'g − fg') / g² |
| Chain | d/dx [f(g(x))] = f'(g(x))·g'(x) |
| Inverse function | If g = f⁻¹, then g'(a) = 1 / f'(g(a)) |
Derivatives of common functions
| Function | Derivative |
|---|---|
| sin x | cos x |
| cos x | −sin x |
| tan x | sec² x |
| cot x | −csc² x |
| sec x | sec x tan x |
| csc x | −csc x cot x |
| eˣ | eˣ |
| aˣ | aˣ ln a |
| ln x | 1/x |
| logₐ x | 1 / (x ln a) |
| arcsin x | 1 / √(1 − x²) |
| arccos x | −1 / √(1 − x²) |
| arctan x | 1 / (1 + x²) |
A quick memory aid: every co-function (cos, cot, csc, arccos) has a negative derivative.
Integral rules
Every indefinite integral needs + C.
| Integral | Result |
|---|---|
| ∫ xⁿ dx (n ≠ −1) | xⁿ⁺¹/(n + 1) + C |
| ∫ 1/x dx | ln abs(x) + C |
| ∫ eˣ dx | eˣ + C |
| ∫ aˣ dx | aˣ / ln a + C |
| ∫ sin x dx | −cos x + C |
| ∫ cos x dx | sin x + C |
| ∫ sec² x dx | tan x + C |
| ∫ csc² x dx | −cot x + C |
| ∫ sec x tan x dx | sec x + C |
| ∫ csc x cot x dx | −csc x + C |
| ∫ tan x dx | −ln abs(cos x) + C |
| ∫ 1/√(1 − x²) dx | arcsin x + C |
| ∫ 1/(1 + x²) dx | arctan x + C |
| ∫ 1/(a² + x²) dx | (1/a) arctan(x/a) + C |
u-substitution checklist
- Choose u as the inside function whose derivative also appears (up to a constant).
- Rewrite every part of the integral in terms of u, including dx.
- For definite integrals, change the limits to u-values, or substitute back before evaluating. Do not mix the two.
The Fundamental Theorem of Calculus
| Part | Formula |
|---|---|
| Evaluation | ∫ₐᵇ f'(x) dx = f(b) − f(a) |
| Accumulation | d/dx ∫ₐˣ f(t) dt = f(x) |
| With a variable upper limit | d/dx ∫ₐ^g(x) f(t) dt = f(g(x))·g'(x) |
| Net change | f(b) = f(a) + ∫ₐᵇ f'(x) dx |
The net change form is the backbone of most rate-in, rate-out free-response questions.
Theorems you must be able to state
| Theorem | Hypotheses | Conclusion |
|---|---|---|
| Intermediate Value Theorem | f continuous on [a, b] | f takes every value between f(a) and f(b) |
| Extreme Value Theorem | f continuous on [a, b] | f has an absolute max and min on [a, b] |
| Mean Value Theorem | f continuous on [a, b], differentiable on (a, b) | Some c in (a, b) has f'(c) = [f(b) − f(a)] / (b − a) |
On FRQs, you must state the hypotheses before you apply the theorem. That is often a point on its own.
Applications of derivatives
| Concept | Formula or test |
|---|---|
| Tangent line at x = a | y = f(a) + f'(a)(x − a) |
| Velocity and acceleration | v(t) = x'(t), a(t) = v'(t) |
| Speed | abs(v(t)); speed increases when v and a have the same sign |
| First Derivative Test | Relative max where f' changes from + to −, min where − to + |
| Second Derivative Test | If f'(c) = 0 and f''(c) < 0, relative max; if f''(c) > 0, relative min |
| Inflection point | f'' changes sign |
Applications of integrals
| Concept | Formula |
|---|---|
| Average value of f on [a, b] | (1/(b − a)) ∫ₐᵇ f(x) dx |
| Displacement | ∫ₐᵇ v(t) dt |
| Total distance | ∫ₐᵇ abs(v(t)) dt |
| Area between curves | ∫ₐᵇ [top − bottom] dx, or ∫ [right − left] dy |
| Volume, disc method | π ∫ₐᵇ R(x)² dx |
| Volume, washer method | π ∫ₐᵇ [R(x)² − r(x)²] dx |
| Volume, known cross sections | ∫ₐᵇ A(x) dx |
Cross-section areas, where s is the side or base length
| Cross section | Area A(x) |
|---|---|
| Square | s² |
| Semicircle (diameter s) | (π/8) s² |
| Equilateral triangle | (√3/4) s² |
| Isosceles right triangle (leg s) | s²/2 |
| Rectangle with height h | s·h |
Differential equations (AB and BC)
| Concept | Formula |
|---|---|
| Exponential growth or decay | If dy/dt = ky, then y = y₀eᵏᵗ |
| Separation of variables | Move all y terms to one side with dy, all x terms with dx, then integrate both sides |
BC-only formulas
Integration and differential equations
| Concept | Formula |
|---|---|
| Integration by parts | ∫ u dv = uv − ∫ v du |
| Euler's method | yₙ₊₁ = yₙ + Δx · (dy/dx at (xₙ, yₙ)) |
| Logistic model | dy/dt = ky(1 − y/L); y approaches L, and grows fastest at y = L/2 |
| Arc length, y = f(x) | ∫ₐᵇ √(1 + [f'(x)]²) dx |
| Improper integral | ∫ₐ^∞ f(x) dx = lim (b→∞) ∫ₐᵇ f(x) dx |
Parametric, polar, and vector-valued functions
| Concept | Formula |
|---|---|
| Parametric slope | dy/dx = (dy/dt) / (dx/dt) |
| Parametric second derivative | d²y/dx² = [d/dt (dy/dx)] / (dx/dt) |
| Speed | √((dx/dt)² + (dy/dt)²) |
| Arc length (parametric) | ∫ √((dx/dt)² + (dy/dt)²) dt |
| Polar to rectangular | x = r cos θ, y = r sin θ |
| Polar area | (1/2) ∫ r² dθ |
Infinite sequences and series
| Concept | Formula |
|---|---|
| Geometric series | Σ arⁿ (n from 0) = a / (1 − r) if abs(r) < 1; diverges otherwise |
| Taylor series about x = a | Σ f⁽ⁿ⁾(a)/n! · (x − a)ⁿ |
| eˣ | 1 + x + x²/2! + x³/3! + … |
| sin x | x − x³/3! + x⁵/5! − … |
| cos x | 1 − x²/2! + x⁴/4! − … |
| 1/(1 − x) | 1 + x + x² + x³ + …, for abs(x) < 1 |
| Alternating series error bound | abs(error) ≤ abs(first omitted term) |
| Lagrange error bound | abs(Rₙ(x)) ≤ max abs(f⁽ⁿ⁺¹⁾(z)) · abs(x − a)ⁿ⁺¹ / (n + 1)! |
You should also know when to apply the nth term test, the integral test, the p-series test, the comparison and limit comparison tests, the alternating series test, and the ratio test, including each test's conditions.
How to actually memorize this
- Derive instead of memorizing where you can. The quotient rule follows from the product rule, and the integral of tan x follows from u-substitution with u = cos x.
- Use active recall. Cover the right column and write the answers from memory, then check.
- Practice in context. Formulas stick when you use them on released free-response questions. Our AP Calculus FRQ tips show how each formula gets scored.
- Review daily in the final weeks. Five minutes a day beats an hour the night before.
To see which units these formulas belong to, read AP Calculus AB units explained. For a full plan, see how to get a 5 on AP Calculus AB, and check practice results with the AP Calculus AB or BC score calculator.
How Ivystar can help
Knowing the formulas is step one; applying them under time pressure is what earns a top score. Ivystar's AP Exam Prep gives you a tutor from schools like Harvard, Penn, Cornell, Stanford, or MIT, with a "Get an A or we work for free" guarantee on your class grade. Book a free 20-minute strategy call to get started.
Frequently asked questions
Do you get a formula sheet on the AP Calculus exam?+
No. Neither AP Calculus AB nor AP Calculus BC provides a formula sheet, so you must memorize derivative rules, integral rules, and key theorems.
What formulas are only on AP Calculus BC?+
BC adds integration by parts, Euler's method, logistic growth, arc length, parametric and polar formulas, and infinite series topics such as Taylor series and error bounds.
What is the most important formula on AP Calculus?+
The Fundamental Theorem of Calculus, especially the net change form f(b) = f(a) + ∫ f'(x) dx, appears in some form on most exams and drives many free-response questions.
How should I memorize AP Calculus formulas?+
Use active recall by covering the answers and writing them from memory, derive formulas where possible, and apply them on released free-response questions so they stick in context.
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