AP Calculus AB Formula Sheet 2026 — Complete Reference Guide
AP Calculus AB does NOT provide a formula sheet on the exam. Everything must be memorized. This guide organizes all the formulas you need — derivatives, integrals, theorems, and key rules — in one reference.
Important: AP Calc AB Has No Formula Sheet
Unlike AP Physics or AP Chemistry, AP Calculus AB does not give you any formulas on exam day. The equation sheet is blank. Everything on this page must be in your memory.
Limits
Limit definition of derivative: $$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$
Key limits to know: $$\lim_{x \to 0} \frac{\sin x}{x} = 1 \qquad \lim_{x \to 0} \frac{1 - \cos x}{x} = 0$$
L'Hôpital's Rule (for 0/0 or ∞/∞ forms): $$\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}$$
Derivatives
Basic Rules
| Rule | Formula |
|---|---|
| Constant | $\frac{d}{dx}[c] = 0$ |
| Power rule | $\frac{d}{dx}[x^n] = nx^{n-1}$ |
| Constant multiple | $\frac{d}{dx}[cf(x)] = cf'(x)$ |
| 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 | $\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$ |
Trigonometric Derivatives
| Function | Derivative |
|---|---|
| $\sin x$ | $\cos x$ |
| $\cos x$ | $-\sin x$ |
| $\tan x$ | $\sec^2 x$ |
| $\cot x$ | $-\csc^2 x$ |
| $\sec x$ | $\sec x \tan x$ |
| $\csc x$ | $-\csc x \cot x$ |
Inverse Trig Derivatives
| Function | Derivative |
|---|---|
| $\arcsin x$ | $\dfrac{1}{\sqrt{1-x^2}}$ |
| $\arccos x$ | $\dfrac{-1}{\sqrt{1-x^2}}$ |
| $\arctan x$ | $\dfrac{1}{1+x^2}$ |
Exponential and Logarithmic Derivatives
| Function | Derivative |
|---|---|
| $e^x$ | $e^x$ |
| $a^x$ | $a^x \ln a$ |
| $\ln x$ | $\dfrac{1}{x}$ |
| $\log_a x$ | $\dfrac{1}{x \ln a}$ |
Integrals
Basic Rules
| Rule | Formula |
|---|---|
| Constant | $\int c,dx = cx + C$ |
| Power rule | $\int x^n,dx = \dfrac{x^{n+1}}{n+1} + C \quad (n \neq -1)$ |
| Reciprocal | $\int \dfrac{1}{x},dx = \ln |
| Exponential | $\int e^x,dx = e^x + C$ |
| $a^x$ | $\int a^x,dx = \dfrac{a^x}{\ln a} + C$ |
Trigonometric Integrals
| Integral | Result |
|---|---|
| $\int \sin x,dx$ | $-\cos x + C$ |
| $\int \cos x,dx$ | $\sin x + C$ |
| $\int \sec^2 x,dx$ | $\tan x + C$ |
| $\int \csc^2 x,dx$ | $-\cot x + C$ |
| $\int \sec x \tan x,dx$ | $\sec x + C$ |
| $\int \csc x \cot x,dx$ | $-\csc x + C$ |
Inverse Trig Integrals
$$\int \frac{1}{\sqrt{1-x^2}},dx = \arcsin x + C$$
$$\int \frac{1}{1+x^2},dx = \arctan x + C$$
Fundamental Theorems of Calculus
FTC Part 1 (Derivative of an integral): $$\frac{d}{dx}\int_a^x f(t),dt = f(x)$$
With chain rule: $$\frac{d}{dx}\int_a^{g(x)} f(t),dt = f(g(x)) \cdot g'(x)$$
FTC Part 2 (Evaluation): $$\int_a^b f(x),dx = F(b) - F(a)$$
where F is any antiderivative of f.
Key Theorems
Mean Value Theorem (MVT): If f is continuous on [a,b] and differentiable on (a,b), then there exists c such that: $$f'(c) = \frac{f(b) - f(a)}{b - a}$$
Intermediate Value Theorem (IVT): If f is continuous on [a,b] and k is between f(a) and f(b), then there exists c in (a,b) with f(c) = k.
Extreme Value Theorem: If f is continuous on [a,b], then f attains both an absolute maximum and absolute minimum.
Rolle's Theorem: Special case of MVT: if f(a) = f(b), then f'(c) = 0 for some c in (a,b).
Applications
Area between two curves: $$A = \int_a^b [f(x) - g(x)],dx \quad \text{(where } f \geq g\text{)}$$
Volume by disk/washer method (rotating about x-axis): $$V = \pi\int_a^b [f(x)]^2,dx \quad \text{(disk)}$$
$$V = \pi\int_a^b \left([f(x)]^2 - [g(x)]^2\right),dx \quad \text{(washer)}$$
Average value of a function: $$f_{avg} = \frac{1}{b-a}\int_a^b f(x),dx$$
Position, velocity, acceleration: $$v(t) = s'(t) \qquad a(t) = v'(t) = s''(t)$$
$$s(t) = s(0) + \int_0^t v(u),du$$
Total distance traveled (not displacement): $$\text{distance} = \int_a^b |v(t)|,dt$$
What to Memorize vs. What to Derive
Must memorize cold:
- All trig derivatives and integrals
- Product rule, quotient rule, chain rule
- FTC Part 1 and 2
- L'Hôpital's Rule
- Area/volume formulas
- MVT and IVT statements
Can derive quickly:
- Quotient rule from product + chain
- Inverse trig derivatives (using implicit differentiation)
- Integration by substitution (from chain rule)
AP Calc AB vs AP Calc BC Formula Comparison
AP Calc BC covers additional topics not in AB:
| BC-Only Topic | Key Formulas |
|---|---|
| Integration by parts | $\int u,dv = uv - \int v,du$ |
| Partial fractions | Decompose rational functions |
| Taylor/Maclaurin series | $f(x) = \sum \frac{f^{(n)}(a)}{n!}(x-a)^n$ |
| Parametric derivatives | $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$ |
| Polar area | $A = \frac{1}{2}\int_\alpha^\beta r^2,d\theta$ |
| Series convergence tests | Ratio, integral, comparison tests |
See the full AP Calc BC Formula Sheet.