Home ›
Cheat Sheets › AP Calculus Cheat Sheet 2026
AP Calculus Cheat Sheet 2026 — Limits, Derivatives & Integrals
Quick reference for AP Calculus AB and BC · Updated for 2026 exam
This cheat sheet covers every essential rule and formula for AP Calculus AB and BC. The College Board does not provide a formula sheet on the AP Calculus exam — you must know these cold.
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 | $\lim_{x \to a} f(x) = f(a)$ |
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 |
|---|
| $\sin x$ | $\cos x$ |
| $\cos x$ | $-\sin x$ |
| $\tan x$ | $\sec^2 x$ |
| $\cot x$ | $-\csc^2 x$ |
| $\sec x$ | $\sec x \tan x$ |
| $\csc x$ | $-\csc x \cot x$ |
| $e^x$ | $e^x$ |
| $a^x$ | $a^x \ln a$ |
| $\ln x$ | $\frac{1}{x}$ |
| $\log_a x$ | $\frac{1}{x \ln a}$ |
| $\arcsin x$ | $\frac{1}{\sqrt{1-x^2}}$ |
| $\arccos x$ | $\frac{-1}{\sqrt{1-x^2}}$ |
| $\arctan x$ | $\frac{1}{1+x^2}$ |
Integral Rules
| Rule | Formula |
|---|
| Power rule | $\int x^n\,dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)$ |
| $\int \frac{1}{x}\,dx$ | $\ln|x| + C$ |
| $\int e^x\,dx$ | $e^x + C$ |
| $\int a^x\,dx$ | $\frac{a^x}{\ln a} + C$ |
| $\int \sin x\,dx$ | $-\cos x + C$ |
| $\int \cos x\,dx$ | $\sin x + C$ |
| $\int \sec^2 x\,dx$ | $\tan x + C$ |
| $\int \sec x \tan x\,dx$ | $\sec x + C$ |
| $\int \frac{1}{\sqrt{1-x^2}}\,dx$ | $\arcsin x + C$ |
| $\int \frac{1}{1+x^2}\,dx$ | $\arctan x + C$ |
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 | $\sum_{n=0}^\infty \frac{(-1)^n x^{2n+1}}{(2n+1)!}$ |
| Parametric derivatives | $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$ |
| Polar area | $A = \frac{1}{2}\int_\alpha^\beta [r(\theta)]^2\,d\theta$ |
Related Resources