AP Calculus Cheat Sheet 2026 — Limits, Derivatives & Integrals
Every essential formula for AP Calculus AB and BC in one place. Use this as a study reference — not a cram sheet the night before.
Limits
| Rule | Formula |
|---|---|
| Definition of derivative | $f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$ |
| L'Hôpital's Rule (0/0 or ∞/∞) | $\lim \frac{f}{g} = \lim \frac{f'}{g'}$ |
| Special limit 1 | $\lim_{x \to 0} \frac{\sin x}{x} = 1$ |
| Special limit 2 | $\lim_{x \to 0} \frac{1-\cos x}{x} = 0$ |
| Continuity requirement | $\lim_{x \to a} f(x) = f(a)$ |
Limit Examples
Example 1: Find $\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$
Factor: $\frac{(x-2)(x+2)}{x-2} = x+2$. Plug in $x=2$: answer is $\mathbf{4}$.
Example 2: Find $\lim_{x \to 0} \frac{\sin(3x)}{x}$
Rewrite as $3 \cdot \frac{\sin(3x)}{3x}$. Since $\lim_{u\to 0}\frac{\sin u}{u}=1$, the answer is $\mathbf{3}$.
Derivative Rules
| Rule | Formula |
|---|---|
| Power rule | $\frac{d}{dx}[x^n] = nx^{n-1}$ |
| Constant multiple | $\frac{d}{dx}[cf] = cf'$ |
| Sum/Difference | $\frac{d}{dx}[f \pm g] = f' \pm g'$ |
| Product rule | $\frac{d}{dx}[fg] = f'g + fg'$ |
| Quotient rule | $\frac{d}{dx}\left[\frac{f}{g}\right] = \frac{f'g - fg'}{g^2}$ |
| Chain rule | $\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$ |
Common Derivatives
| Function | Derivative |
|---|---|
| $\sin x$ | $\cos x$ |
| $\cos x$ | $-\sin x$ |
| $\tan x$ | $\sec^2 x$ |
| $\cot x$ | $-\csc^2 x$ |
| $\sec x$ | $\sec x \tan x$ |
| $\csc x$ | $-\csc x \cot x$ |
| $e^x$ | $e^x$ |
| $a^x$ | $a^x \ln a$ |
| $\ln x$ | $\frac{1}{x}$ |
| $\log_a x$ | $\frac{1}{x \ln a}$ |
| $\arcsin x$ | $\frac{1}{\sqrt{1-x^2}}$ |
| $\arccos x$ | $\frac{-1}{\sqrt{1-x^2}}$ |
| $\arctan x$ | $\frac{1}{1+x^2}$ |
Derivative Examples
Chain Rule: Differentiate $f(x) = \sin(x^2)$
$f'(x) = \cos(x^2) \cdot 2x = \mathbf{2x\cos(x^2)}$
Product Rule: Differentiate $g(x) = x^2 e^x$
$g'(x) = 2x \cdot e^x + x^2 \cdot e^x = \mathbf{e^x(2x + x^2)}$
Integral Rules
| Rule | Formula |
|---|---|
| Power rule | $\int x^n\,dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)$ |
| $\int \frac{1}{x}\,dx$ | $\ln|x| + C$ |
| $\int e^x\,dx$ | $e^x + C$ |
| $\int a^x\,dx$ | $\frac{a^x}{\ln a} + C$ |
| $\int \sin x\,dx$ | $-\cos x + C$ |
| $\int \cos x\,dx$ | $\sin x + C$ |
| $\int \sec^2 x\,dx$ | $\tan x + C$ |
| $\int \sec x \tan x\,dx$ | $\sec x + C$ |
| $\int \frac{1}{\sqrt{1-x^2}}\,dx$ | $\arcsin x + C$ |
| $\int \frac{1}{1+x^2}\,dx$ | $\arctan x + C$ |
Integral Examples
Definite integral: Evaluate $\int_0^2 x^3\,dx$
$= \left[\frac{x^4}{4}\right]_0^2 = \frac{16}{4} - 0 = \mathbf{4}$
U-substitution: Evaluate $\int 2x\cos(x^2)\,dx$
Let $u = x^2$, $du = 2x\,dx$. Then $\int \cos u\,du = \sin u + C = \mathbf{\sin(x^2) + C}$.
Fundamental Theorem of Calculus
FTC Part 1: $\frac{d}{dx}\int_a^x f(t)\,dt = f(x)$
FTC Part 2: $\int_a^b f(x)\,dx = F(b) - F(a)$ where $F' = f$
Chain rule extension: $\frac{d}{dx}\int_a^{g(x)} f(t)\,dt = f(g(x)) \cdot g'(x)$
Applications of Derivatives
| Concept | Key Rule |
|---|---|
| Critical points | $f'(x) = 0$ or undefined |
| Increasing | $f'(x) > 0$ |
| Decreasing | $f'(x) < 0$ |
| Concave up | $f''(x) > 0$ |
| Concave down | $f''(x) < 0$ |
| Inflection point | $f''(x)$ changes sign |
| Local max (1st deriv test) | $f'$ changes + to − |
| Local min (1st deriv test) | $f'$ changes − to + |
| Mean Value Theorem | $f'(c) = \frac{f(b)-f(a)}{b-a}$ for some $c \in (a,b)$ |
Applications of Integrals
| Application | Formula |
|---|---|
| Area between curves | $\int_a^b [f(x) - g(x)]\,dx$ where $f \geq g$ |
| Disk method (rotation about x-axis) | $V = \pi\int_a^b [f(x)]^2\,dx$ |
| Washer method | $V = \pi\int_a^b \left([R(x)]^2 - [r(x)]^2\right)dx$ |
| Average value of f on [a,b] | $\frac{1}{b-a}\int_a^b f(x)\,dx$ |
| Position from velocity | $s(t) = s(0) + \int_0^t v(\tau)\,d\tau$ |
| Net displacement | $\int_a^b v(t)\,dt$ |
| Total distance | $\int_a^b |v(t)|\,dt$ |
BC-Only Topics
| Topic | Key Formula |
|---|---|
| Integration by parts | $\int u\,dv = uv - \int v\,du$ |
| Arc length | $L = \int_a^b \sqrt{1+[f'(x)]^2}\,dx$ |
| Logistic differential equation | $\frac{dP}{dt} = kP\left(1 - \frac{P}{M}\right)$ |
| Taylor series (center $a$) | $\sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!}(x-a)^n$ |
| $e^x$ Maclaurin series | $\sum_{n=0}^\infty \frac{x^n}{n!}$ |
| $\sin x$ Maclaurin series | $\sum_{n=0}^\infty \frac{(-1)^n x^{2n+1}}{(2n+1)!}$ |
| $\cos x$ Maclaurin series | $\sum_{n=0}^\infty \frac{(-1)^n x^{2n}}{(2n)!}$ |
| Parametric derivatives | $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$ |
| Polar area | $A = \frac{1}{2}\int_\alpha^\beta [r(\theta)]^2\,d\theta$ |
Common AP Exam Tasks
- Find a derivative — Apply chain, product, or quotient rule. Know all trig and inverse trig derivatives without hesitation.
- Analyze a function — Use $f'$ for increasing/decreasing intervals, $f''$ for concavity. Identify critical points and inflection points.
- Evaluate a definite integral — Apply FTC Part 2: find the antiderivative, plug in bounds, subtract. Show each step.
- Solve a related rates problem — Differentiate implicitly with respect to $t$. Label all variables and rates before solving.
- Interpret an accumulation function — If $g(x) = \int_a^x f(t)\,dt$, then $g'(x) = f(x)$ by FTC Part 1. Know this cold.
- Apply MVT or IVT — MVT requires differentiability on $(a,b)$ and continuity on $[a,b]$. Always verify conditions before citing the theorem.
- BC: Series convergence — Know the ratio test and the four standard Maclaurin series ($e^x$, $\sin x$, $\cos x$, $\frac{1}{1-x}$).
How to Memorize These Formulas Fast
- Group by pattern — Trig derivatives follow pairs: $\sin \leftrightarrow \cos$, with co-functions picking up a negative. Learn the pattern, not 12 separate facts.
- Work problems, don't just read — Do 5 chain-rule problems back to back until it's automatic, then move to the next rule. Reading alone does not build recall.
- Write formulas by hand — Handwriting activates different memory pathways than typing. Write each formula 3 times before a study session.
- Condense to one page — Take this full cheat sheet and reduce it to a single handwritten page. The act of selecting what matters forces active recall.
- Daily 10-minute review — Cover the formula column and try to reproduce each one from memory. Ten minutes every day beats a two-hour cram session.
Frequently Asked Questions
Do you get a formula sheet on AP Calculus AB?
No. The College Board provides no formula sheet for AP Calculus AB or BC. Every formula on this page must be memorized before exam day.
What formulas are most important for AP Calc AB?
Prioritize: chain rule, product rule, quotient rule, FTC Part 1 and 2, and the definition of derivative. These appear on almost every FRQ. Also know all trig derivatives and the power rule for integrals.
Is AP Calculus AB enough to get a 5?
Yes. The curve is generous: roughly 62–65% of total points earns a 5. Master derivatives, integrals, and the FTC, and you are in a strong position.
What is the difference between AP Calc AB and BC formulas?
BC covers everything in AB plus: integration by parts, Taylor/Maclaurin series, parametric equations, polar coordinates, and logistic differential equations. See the BC-Only section above.