Introduction
Here’s a number that might surprise you: the College Board reports that only about 60% of AP Calculus AB students score a 3 or higher on the exam each year. For BC, the pass rate is better — but the content is significantly harder. One of the biggest reasons students struggle? They simply don’t have the formulas memorized when it counts most.
I’ve seen it happen over and over again. A student understands the concepts perfectly during practice, but then freezes during the timed exam because they can’t recall the Chain Rule or the formula for arc length. That’s where this AP Calculus formula cheat sheet comes in.
This guide is organized by topic — not randomly dumped into one overwhelming list. Whether you’re cramming the night before the exam or building a systematic review over several weeks, you’ll find every essential formula right here, clearly explained and logically grouped.
Bookmark this page. Print it out. Write it on flashcards. Whatever it takes — because knowing these formulas cold is the single fastest way to boost your AP Calculus score.
How to Use This AP Calculus Formula Cheat Sheet
Before diving into the formulas themselves, let’s set you up for success. This cheat sheet covers both AP Calculus AB and AP Calculus BC. Formulas that are BC-only will be clearly labeled throughout.
| 📌 Quick Tips for Using This SheetStudy one section at a time — don’t try to memorize everything at onceWrite formulas by hand; the physical act strengthens memory retentionTest yourself: cover the right column and try to recall each formulaFocus extra time on formulas you consistently forget — not the ones you already knowOn exam day, the AP exam does NOT provide a formula sheet — you must know these cold |
Limits Formulas
Limits are the foundation of calculus. Everything — derivatives, integrals, continuity — is built on the concept of a limit. You don’t need to memorize every limit theorem, but the following are tested frequently on the AP exam.
Core Limit Properties
Let \(\lim_{x \to a} f(x) = L\) and \(\lim_{x \to a} g(x) =M \).
| Limit of a constant | $$\lim_{x \to a} c = c$$ |
| Sum/Difference Rule | $$\lim_{x \to a} [f(x) ± g(x)] = L ± M$$ |
| Product Rule | $$\lim_{x \to a} [f(x) · g(x)] = L · M$$ |
| Quotient Rule | $$\lim_{x \to a} \frac{f(x)}{ g(x)} = \frac{L}{ M}, \, (M ≠ 0)$$ |
Special Trigonometric Limits
| Fundamental trig limit | $$\lim_{x \to 0} \frac{\sin x}{x}= 1$$ |
| Second trig limit | $$\lim_{x \to 0} \frac{1 – \cos x}{x} = 0$$ |
L’Hôpital’s Rule
| L’Hôpital’s Rule | $$\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}\, [\frac{0}{0} or \frac{\infty}{\infty} \mbox{ form}]$$ |
Use L’Hôpital’s Rule when direct substitution produces an indeterminate form like 0/0 or ∞/∞. Differentiate numerator and denominator separately, then re-evaluate the limit.
Derivative Formulas
Derivatives measure rates of change. The AP exam tests derivative rules heavily — expect multiple free-response and multiple-choice questions that require fluency with these formulas.
Basic Differentiation Rules
| Rule | Formula |
| Power Rule | $$\frac{d}{dx} x^n = n·x^{n-1}$$ |
| Constant Rule | $$\frac{d}{dx} c = 0$$ |
| Constant Multiple | $$\frac{d}{dx} [c·f(x)] = c·f'(x)$$ |
| Sum / Difference | $$\frac{d}{dx} [f ± g] = f’ ± g’$$ |
| Product Rule | $$\frac{d}{dx} [f·g] = f’g + fg’$$ |
| Quotient Rule | $$\frac{d}{dx} \frac{f}{g} = \frac{f’g − fg’}{ g^2}$$ |
| Chain Rule | $$\frac{d}{dx} f(g(x)) = f'(g(x))·g'(x)$$ |
Derivatives of Trigonometric Functions
| Function | Derivative |
| $$\sin x$$ | $$\cos x$$ |
| $$\cos x$$ | $$−\sin x$$ |
| $$\tan x$$ | $$\sec^2 x$$ |
| $$\cot x$$ | $$−\csc^2 x$$ |
| $$\sec x$$ | $$\sec x · \tan x$$ |
| $$\csc x$$ | $$−\csc x · \cot x$$ |
Derivatives of Exponential and Logarithmic Functions
| Function | Derivative |
| $$e^x$$ | $$e^x$$ |
| $$a^x$$ | $$a^x · \ln a$$ |
| $$\ln x$$ | $$\frac{1}{ x}$$ |
| $$\log_a x$$ | $$\frac{1 }{ x · \ln a}$$ |
Derivatives of Inverse Trigonometric Functions
|
Function |
Derivative |
|
$$\arcsin x$$ |
\frac{1}{\sqrt{1 − x^2}} |
|
$$\arccos x$$ |
−\frac{1 }{ \sqrt{1 − x^2}} |
|
$$\arctan x$$ |
\frac{1 }{ 1 + x^2} |
| 💡 Exam Tip: Chain Rule PitfallThe Chain Rule is one of the most commonly missed formulas under exam pressure.Always ask yourself: ‘Is there a function inside another function?’ If yes — use the Chain Rule.Example: $$\frac{d}{dx} \sin(3x^2)= \cos(3x^2) · 6x$$ |
Integral Formulas
Integration is the reverse of differentiation. AP Calculus AB covers basic integration techniques; AP Calculus BC adds integration by parts, partial fractions, and improper integrals. Know both if you’re taking BC!
Basic Integration Rules
| Rule | Formula |
| Power Rule | $$\int x^n dx = \frac{x^{n+1}}{n+1} + C, \, (n \neq −1)$$ |
| Constant Multiple | $$\int c·f(x) dx = c · \int f(x) dx$$ |
| Sum / Difference | $$\int [f \pm g] dx = \int f dx \pm \int g dx$$ |
| $$\int \frac{1}{x} dx$$ | $$\ln |x| + C$$ |
| $$ \int e^x dx$$ | $$e^x + C$$ |
| $$\int a^x dx$$ | $$\frac{a^x}{\ln a} + C$$ |
Integrals of Trigonometric Functions
| Integral | Result |
| $$\int \sin x dx$$ | $$−\cos x + C$$ |
| $$\int \cos x dx$$ | $$\sin x + C$$ |
| $$\int \tan x dx$$ | $$−\ln |\cos x| + C = \ln |\sec x| + C$$ |
| $$ \cot x dx$$ | $$\ln |\sin x| + C$$ |
| $$\int \sec x dx$$ | $$\ln |\sec x + \tan x| + C$$ |
| $$\int \csc x dx$$ | $$−\ln |\csc x + \cot x| + C$$ |
| $$\int \sec^2 x dx$$ | $$\tan x + C$$ |
| $$\int \csc^2 x dx$$ | $$−\cot x + C$$ |
| $$\int \sec x \tan x dx$$ | $$\sec x + C$$ |
| $$\int \csc x \cot x dx$$ | $$−\csc x + C$$ |
Integrals of Inverse Trig Forms
|
Integral |
Result |
|
\int \frac{1}{\sqrt{1−x^2}} dx |
$$\arcsin x + C$$ |
|
\int \frac{1}{\sqrt{1+x^2} }dx |
$$\arctan x + C$$ |
Integration by Parts (BC Only)
| Integration by Parts | $$\int u dv = uv − \int v du$$ |
Tip — LIATE Rule: Choose u in this priority order: Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential. This gives you the best chance that ∫ v du is easier than the original integral.
The Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus (FTC) is arguably the most important result in all of AP Calculus. It links differentiation and integration — two operations that seem completely separate — into one unified framework.
| FTC Part 1 | $$\frac{d}{dx} \int_a^x f(t) dt = f(x)$$ |
| FTC Part 2 | $$\int_a^b f(x) dx = F(b) − F(a), $$ $$\mbox{ where } F'(x) = f(x)$$ |
Part 1 tells you that differentiation and integration undo each other. Part 2 gives you the practical method for evaluating definite integrals — find the antiderivative, plug in the bounds, subtract.
| 🔑 FTC on the AP ExamFTC Part 1 frequently appears in free-response questions as d/dx of an integral with a variable upper boundWhen the upper limit is a function of x (not just x), apply the Chain Rule: $$\frac{d}{dx} \int_a^{g(x)} f(t) dt = f(g(x)) · g'(x)$$ FTC Part 2 is used in virtually every definite integral evaluation question |
Applications of Derivatives
Knowing how to differentiate is only half the battle. The AP exam also tests whether you can apply derivatives to solve real problems. These are the formulas and relationships you need.
| Application | Formula / Concept |
| Equation of tangent line at \((a, f(a))\) | \(y − f(a) = f'(a)(x − a)\) |
| Mean Value Theorem (MVT) | \(f'(c) = \frac{f(b)−f(a)}{ b−a}\) for some \(c\) in \((a,b)\) |
| Critical point condition | \(f'(x) = 0\) or \(f'(x)\) is undefined |
| Relative max (1st derivative test) | \(f’\) changes from + to − |
| Relative min (1st derivative test) | \(f’\) changes from − to + |
| Concave up | f''(x) > 0 |
| Concave down | f''(x) < 0 |
| Inflection point | f'' changes sign |
| Absolute max/min on [a,b] | Compare \(f\) at critical pts and endpoints |
| Related rates | Differentiate both sides with respect to \(t\) |
| Linearization (local linear approx.) | \(L(x) = f(a) + f'(a)(x − a)\) |
Applications of Integrals
Definite integrals are used to compute area, volume, average value, and accumulated change. These applications appear heavily on both the AB and BC free-response sections.
| Application | Formula |
| Area between two curves | \int_a^b [f(x) − g(x)] dx (\(f \geq g\) on \([a,b]\)) |
| Average value of f on [a,b] | \frac{1}{b−a} · \int_a^b f(x) dx |
| Disk method (x-axis) | \pi · \int_a^b [f(x)]^2 dx |
| Washer method | \pi · \int_a^b ([f(x)]^2 − [g(x)]^2) dx |
| Shell method (BC) | 2\pi · \int_a^b x · f(x) dx |
| Arc length (BC) | \int_a^b \sqrt{1 + [f'(x)]^2} dx |
| Distance traveled | \int_a^b |v(t)| dt |
| Displacement | \int_a^b v(t) dt |
| Net change | \int_a^b f'(x) dx = f(b) − f(a) |
BC-Only Formulas: Series, Parametric, and Polar
If you’re taking AP Calculus BC, the following formulas represent a significant portion of the additional material. Don’t skip this section — the BC exam will test all of it.
Sequences and Series
| Series / Test | Formula or Condition |
| Geometric series sum | S = \frac{a}{1−r}, |r| < 1 |
| Taylor series (centered at a) | \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} · (x−a)^n |
| Maclaurin series for eˣ | \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + ... |
| Maclaurin series for sin x | \sum_{n=0}^{\infty} \frac{(−1)^n x^{2n+1}}{(2n+1)!} |
| Maclaurin series for cos x | \sum_{n=0}^{\infty} \frac{(−1)^n x^{2n}}{(2n)!} |
| Maclaurin series for 1/(1−x) | \sum_{n=0}^{\infty} x^n = 1 + x + x^2 + x^3 + ... \(|x| < 1 \) |
| Alternating Series Remainder | \(|\mbox{Error}| ≤ |a_{n+1}|\) |
| Lagrange Error Bound | \(|R_n(x)| ≤ M · \frac{|x−a|^{n+1}}{(n+1)!}\) |
Parametric Equations (BC)
| dy/dx (parametric) | \(\frac{dy}{dx} = \frac{dy}{dt} / \frac{dx}{dt}\) |
| d²y/dx² (parametric) | \(\frac{d²^2y}{dx^2} = [\frac{d}{dt}(\frac{dy}{dx})] / (\frac{dx}{dt}) \) |
| Arc length (parametric) | \int \sqrt{(\frac{dx}{dt})^2+ (\frac{dy}{dt})^2}dt |
Polar Coordinates (BC)
| Area in polar | A = \frac{1}{2} \int_a^b [r(θ)]^2 dθ |
| Arc length in polar | \int_a^b \sqrt{r^2 + (\frac{dr}{dθ})^2} dθ |
| \(\frac{dy}{dx}\) in polar | $$\frac{dy}{dx} = \frac{r’ sin θ + r cos θ}{r’ cos θ − r sin θ}$$ |
Master Formula Quick-Reference Table
Use this table as a final review before exam day. If you can look at the left column and recall the right column without hesitation, you’re ready.
| Topic | Must-Know Formula |
| Power Rule (deriv.) | d/dx [xⁿ] = n·xⁿ⁻¹ |
| Chain Rule | d/dx [f(g(x))] = f'(g(x))·g'(x) |
| Product Rule | d/dx [fg] = f’g + fg’ |
| Quotient Rule | d/dx [f/g] = (f’g − fg’) / g² |
| FTC Part 2 | ∫ₐᵇ f dx = F(b) − F(a) |
| Power Rule (integ.) | ∫xⁿ dx = xⁿ⁺¹/(n+1) + C |
| ∫ sin x | −cos x + C |
| ∫ cos x | sin x + C |
| ∫ eˣ | eˣ + C |
| ∫ 1/x | ln|x| + C |
| Integration by Parts (BC) | ∫ u dv = uv − ∫ v du |
| MVT | f'(c) = [f(b)−f(a)]/(b−a) |
| Average value | (1/(b−a)) ∫ₐᵇ f dx |
| Area between curves | ∫ₐᵇ [f−g] dx |
| Disk method | π ∫ₐᵇ [f(x)]² dx |
| Geometric series (BC) | S = a/(1−r), |r|<1 |
| Taylor series (BC) | Σ f⁽ⁿ⁾(a)/n! · (x−a)ⁿ |
| Polar area (BC) | (1/2)∫[r(θ)]² dθ |
Conclusion
Formula mastery is exam mastery. That’s the bottom line. The AP Calculus exam does not give you a reference sheet — every single formula above must come from memory when you’re sitting in that exam room under the clock.
The good news? You’ve got everything you need right here. This AP Calculus formula cheat sheet covers every major category — from limits and derivatives to integrals, FTC applications, and BC-exclusive topics like Taylor series and polar coordinates. There are no surprises if you know this material cold.
Here’s your action plan: review one section per day, write out the formulas by hand, and quiz yourself until recall is instant. If you haven’t already, pair this cheat sheet with our AP Calculus 4-Week Study Schedule and our AP Calculus 2-Week Cram Plan for a complete exam-prep system.
You’ve put in the work. Now make sure the formulas are locked in — because on exam day, the student who knows their formulas owns the exam. Good luck!