AP Calculus AB/BC Formula Sheet

What You Need to Know

This “formula sheet” is the College Board reference you get during the AP Calculus AB/BC exam. The big win: it reminds you of standard derivatives, antiderivatives, and (for BC) key series—so you can focus on setup/reasoning instead of memorization.

Two rules of thumb:

The sheet mainly covers core derivative/integral identities + FTC + average value (and BC series).
You still must know how/when to use them (chain rule, substitutions, interpreting integrals, etc.).

Critical reminder: The sheet gives formulas, not judgment. On FRQs, points often come from the setup (correct expression, bounds, units/interpretation), not just the final number.

What’s “core” on the sheet

Derivative definition and derivative rules
Common derivatives (powers, exponentials/logs, trig, inverse trig)
Common antiderivatives (matching the derivative list)
Fundamental Theorem of Calculus (both directions)
Average value of a function on an interval
BC only: geometric series + Taylor/Maclaurin series facts and standard Maclaurin series

Step-by-Step Breakdown

Use this process whenever you’re “mining” the formula sheet during a problem.

A. Derivative problems (rates, tangents, optimization, related rates)

Identify the function type (power, trig, exponential/log, composition, product/quotient).
Decide which rule(s) you need:
- Composition chain rule
- Product/quotient present product/quotient rule
- Implicit relation differentiate both sides + solve for $\frac{dy}{dx}$
Pull the base derivative from the sheet, then apply your rule(s).
Substitute the evaluation point last (avoid algebra mistakes).

Micro-example (chain + trig):

Find $\frac{d}{dx}\big(\sin(3x^2)\big)$
Base: $\frac{d}{dx}(\sin u)=\cos u$
Chain: $u=3x^2\Rightarrow u'=6x$
Result: $\frac{d}{dx}\big(\sin(3x^2)\big)=\cos(3x^2)\cdot 6x$

B. Definite integrals (accumulation, area, net change)

Interpret what the integral means (signed area? total accumulation? average?).
Choose the right tool:
- If it’s literally $\int_a^b f(x)\,dx$ and you have an antiderivative: use FTC.
- If it’s an accumulation function like $g(x)=\int_a^x f(t)\,dt$ : use FTC Part 1.
Match integrand to a known antiderivative from the sheet (or rewrite to match).
Evaluate carefully with bounds; keep signs straight.

Micro-example (FTC Part 1):

If $g(x)=\int_2^x \sqrt{1+t^2}\,dt$ , then $g'(x)=\sqrt{1+x^2}$ .

C. BC series (Taylor/Maclaurin)

Recognize the target: are you approximating a function near a point, or building a series from known ones?
Start from a known Maclaurin series (from the sheet) or the general Taylor formula.
Transform using algebra, substitution, differentiation, or integration of series.
State interval of convergence (at least from the parent series), then adjust for transformations.

Micro-example (build from geometric):

From $\frac{1}{1-x}=\sum_{n=0}^{\infty} x^n$ for $|x|<1$ ,
Replace $x\to -x$ to get $\frac{1}{1+x}=\sum_{n=0}^{\infty} (-1)^n x^n$ for $|x|<1$ .

Key Formulas, Rules & Facts

Derivative definition + rules

Item	Formula	Notes
Derivative definition	$f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$	Use for first-principles or to justify differentiability ideas.
Constant multiple	$\frac{d}{dx}\big(c\,f(x)\big)=c\,f'(x)$	Pull constants out.
Sum/difference	$\frac{d}{dx}\big(f(x)\pm g(x)\big)=f'(x)\pm g'(x)$	Term-by-term.
Product rule	$\frac{d}{dx}\big(f(x)g(x)\big)=f(x)g'(x)+g(x)f'(x)$	“Left stays, right derives + right stays, left derives.”
Quotient rule	$\frac{d}{dx}\Big(\frac{f(x)}{g(x)}\Big)=\frac{g(x)f'(x)-f(x)g'(x)}{\big(g(x)\big)^2}$	Denominator squared.
Chain rule	$\frac{d}{dx}\big(f(g(x))\big)=f'(g(x))\,g'(x)$	The most-tested rule.

Common derivatives (the “lookup” list)

Function	Derivative	Notes
$x^n$	$\frac{d}{dx}(x^n)=n x^{n-1}$	Works for many real $n$ in AP contexts.
$e^x$	$\frac{d}{dx}(e^x)=e^x$	The “self-derivative.”
$a^x$	$\frac{d}{dx}(a^x)=a^x\ln(a)$	Assume $a>0$ and $a\neq 1$ .
$\ln(x)$	$\frac{d}{dx}(\ln x)=\frac{1}{x}$	Domain $x>0$ . For $\ln\|x\|$ , derivative is still $\frac{1}{x}$ for $x\neq 0$ .
$\log_a(x)$	$\frac{d}{dx}(\log_a x)=\frac{1}{x\ln(a)}$	Often rewrite using natural log.
$\sin x$	$\cos x$
$\cos x$	$-\sin x$
$\tan x$	$\sec^2 x$
$\cot x$	$-\csc^2 x$
$\sec x$	$\sec x\tan x$
$\csc x$	$-\csc x\cot x$
$\arcsin x$	$\frac{1}{\sqrt{1-x^2}}$	Domain note: $\|x\|<1$ for derivative expression.
$\arccos x$	$-\frac{1}{\sqrt{1-x^2}}$	Same denominator as arcsin, negative sign.
$\arctan x$	$\frac{1}{1+x^2}$	Very common with integrals too.

Common antiderivatives (indefinite integrals)

Integrand	Antiderivative	Notes
$x^n$	$\int x^n\,dx=\frac{x^{n+1}}{n+1}+C$	شرط: $n\neq -1$ .
$\frac{1}{x}$	$\int \frac{1}{x}\,dx=\ln\|x\|+C$	Absolute value matters.
$e^x$	$\int e^x\,dx=e^x+C$
$a^x$	$\int a^x\,dx=\frac{a^x}{\ln(a)}+C$
$\sin x$	$\int \sin x\,dx=-\cos x+C$
$\cos x$	$\int \cos x\,dx=\sin x+C$
$\sec^2 x$	$\int \sec^2 x\,dx=\tan x+C$
$\csc^2 x$	$\int \csc^2 x\,dx=-\cot x+C$
$\sec x\tan x$	$\int \sec x\tan x\,dx=\sec x+C$
$\csc x\cot x$	$\int \csc x\cot x\,dx=-\csc x+C$
$\frac{1}{1+x^2}$	$\int \frac{1}{1+x^2}\,dx=\arctan x+C$
$\frac{1}{\sqrt{1-x^2}}$	$\int \frac{1}{\sqrt{1-x^2}}\,dx=\arcsin x+C$

Definite integrals + FTC + properties

Concept	Formula	Notes
Definite integral as limit	$\int_a^b f(x)\,dx=\lim_{n\to\infty}\sum_{i=1}^{n} f(x_i^*)\,\Delta x$	Conceptual foundation for accumulation/Riemann sums.
FTC Part 1	If $g(x)=\int_a^x f(t)\,dt$ then $g'(x)=f(x)$	Also works with chain: $\frac{d}{dx}\int_a^{u(x)} f(t)\,dt=f(u(x))\,u'(x)$ .
FTC Part 2	If $F'(x)=f(x)$ then $\int_a^b f(x)\,dx=F(b)-F(a)$	Your main evaluation tool.
Linearity	$\int_a^b \big(cf(x)\pm g(x)\big)\,dx=c\int_a^b f(x)\,dx\pm \int_a^b g(x)\,dx$	Split and factor constants.
Reverse limits	$\int_b^a f(x)\,dx=-\int_a^b f(x)\,dx$	Fast sign check.
Additivity	$\int_a^b f(x)\,dx=\int_a^c f(x)\,dx+\int_c^b f(x)\,dx$	Break at convenient points.
Average value	$f_{\text{avg}}=\frac{1}{b-a}\int_a^b f(x)\,dx$	Often appears with “mean value” wording.

BC-only: geometric series + Taylor/Maclaurin

Item	Formula	Notes
Geometric series	$\sum_{n=0}^{\infty} ar^n=\frac{a}{1-r}$	Converges for $\|r\|<1$ .
Taylor series (general)	$f(x)=\sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n$	Also defines Taylor polynomial by truncation.
Maclaurin for $e^x$	$e^x=\sum_{n=0}^{\infty}\frac{x^n}{n!}$	Converges for all $x$ .
Maclaurin for $\sin x$	$\sin x=\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n+1}}{(2n+1)!}$	Alternating odd powers.
Maclaurin for $\cos x$	$\cos x=\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n}}{(2n)!}$	Alternating even powers.
Geometric power series	$\frac{1}{1-x}=\sum_{n=0}^{\infty} x^n$	Converges for $\|x\|<1$ .
Maclaurin for $\ln(1+x)$	$\ln(1+x)=\sum_{n=1}^{\infty}(-1)^{n-1}\frac{x^n}{n}$	Converges for $-1<x\le 1$ .
Maclaurin for $\arctan x$	$\arctan x=\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n+1}}{2n+1}$	Converges for $\|x\|\le 1$ .

Examples & Applications

1) Product vs chain (don’t mix them)

Differentiate $h(x)=x^2\cos(5x)$ .

Product rule: $h'(x)=2x\cos(5x)+x^2\cdot \frac{d}{dx}(\cos(5x))$
Chain on cosine: $\frac{d}{dx}(\cos(5x))=-\sin(5x)\cdot 5$
Final: $h'(x)=2x\cos(5x)-5x^2\sin(5x)$

2) FTC Part 1 with a variable upper limit (chain rule version)

Compute $\frac{d}{dx}\left(\int_{0}^{x^3} \sqrt{1+t^2}\,dt\right)$ .

Recognize: derivative of an accumulation function.
Apply FTC Part 1 + chain:
$\frac{d}{dx}\int_0^{u(x)} f(t)\,dt=f(u(x))u'(x)$
Here $f(t)=\sqrt{1+t^2}$ and $u(x)=x^3$ so $u'(x)=3x^2$ .
Result: $\sqrt{1+(x^3)^2}\cdot 3x^2=3x^2\sqrt{1+x^6}$

3) Average value setup (FRQ favorite)

Find the average value of $f(x)=\sin x$ on $[0,\pi]$ .

Use $f_{\text{avg}}=\frac{1}{b-a}\int_a^b f(x)\,dx$ :
$f_{\text{avg}}=\frac{1}{\pi-0}\int_0^{\pi} \sin x\,dx$
Antiderivative: $\int \sin x\,dx=-\cos x$
Evaluate: $\frac{1}{\pi}\Big[-\cos x\Big]_0^{\pi}=\frac{1}{\pi}\big((-\cos\pi)-(-\cos 0)\big)=\frac{1}{\pi}(1+1)=\frac{2}{\pi}$

4) BC series: build a new series quickly

Find a power series for $\frac{1}{1+x^2}$ around $x=0$ and its convergence interval.

Start from $\frac{1}{1-x}=\sum_{n=0}^{\infty} x^n$ for $|x|<1$ .
Substitute $x\to -x^2$ :
$\frac{1}{1-(-x^2)}=\frac{1}{1+x^2}=\sum_{n=0}^{\infty}(-x^2)^n=\sum_{n=0}^{\infty} (-1)^n x^{2n}$
Convergence from $|-x^2|<1\Rightarrow x^2<1\Rightarrow |x|<1$ .

Common Mistakes & Traps

Forgetting the chain rule: You write $\frac{d}{dx}(\sin(3x))=\cos(3x)$ (missing the $\cdot 3$ ).
Fix: Always multiply by derivative of the “inside”: $\cos(3x)\cdot 3$ .
Quotient rule sign flip: Using $\frac{f'g-fg'}{g^2}$ but swapping order randomly.
Fix: Commit to one pattern: $\frac{g f' - f g'}{g^2}$ .
Dropping absolute value in $\ln|x|$ : Writing $\int \frac{1}{x}\,dx=\ln x + C$ .
Why wrong: Derivative of $\ln x$ only works for $x>0$ .
Fix: Use $\ln|x|+C$ .
Mixing up inverse trig derivatives: Confusing arcsin vs arctan.
Fix: Remember the “shapes”: arcsin/arccos have $\sqrt{1-x^2}$ ; arctan has $1+x^2$ .
FTC Part 1 without chain: For $\frac{d}{dx}\int_a^{x^2} f(t)dt$ you answer $f(x^2)$ but miss $\cdot 2x$ .
Fix: Upper limit $u(x)$ means multiply by $u'(x)$ .
Sign errors with trig antiderivatives: Saying $\int \sin x\,dx=\cos x + C$ .
Fix: Differentiate your result quickly: derivative of $-\cos x$ is $\sin x$ .
Power rule exception at $n=-1$ : Trying $\int x^{-1}dx=\frac{x^0}{0}$ .
Fix: Memorize the special case: $\int \frac{1}{x}dx=\ln|x|+C$ .
Series interval endpoints ignored (BC): You state “converges for $|x|<1$ ” but the transformed series actually includes endpoints (like arctan).
Fix: Check endpoints separately when the parent interval is $|x|\le 1$ or when you transform a known boundary.

Memory Aids & Quick Tricks

Trick / mnemonic	Helps you remember	When to use
Trig derivative cycle	$\sin\to\cos\to-\sin\to-\cos\to\sin$	Rapid derivatives of $\sin x$ and $\cos x$ .
“Arcsin has a root, arctan has a plus”	$\frac{d}{dx}(\arcsin x)=\frac{1}{\sqrt{1-x^2}}$ vs $\frac{d}{dx}(\arctan x)=\frac{1}{1+x^2}$	Inverse trig derivatives/integrals.
“Flip the sign, flip the limit”	$\int_b^a f=-\int_a^b f$	Quick checks on definite integrals.
Geometric template	$\frac{1}{1-r}=\sum_{n=0}^{\infty} r^n$	Build many BC power series by substitution.
Differentiate/integrate series term-by-term	$\sum c_n x^n \Rightarrow \sum n c_n x^{n-1}$	Creating new series from known ones (BC).

Quick Review Checklist

You can write derivative rules from the sheet: sum, constant multiple, product, quotient, chain.
You can instantly recall the “big 4” derivatives: $x^n$ , $e^x$ , $\ln x$ , trig.
You remember which inverse trig goes with $\sqrt{1-x^2}$ vs $1+x^2$ .
You can match integrands to antiderivatives (and include $+C$ when appropriate).
You can use FTC both ways:
- $\frac{d}{dx}\int_a^x f(t)dt=f(x)$
- $\int_a^b f(x)dx=F(b)-F(a)$
You can set up average value correctly: $\frac{1}{b-a}\int_a^b f(x)dx$ .
(BC) You know the standard Maclaurin series and the geometric series condition $|r|<1$ .
You check common traps: missing chain factors, wrong signs, missing $\ln|x|$ .

You’ve got this—use the sheet to save memory, and spend your brainpower on setup and interpretation.