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|
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| |
| \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| |
| 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| |
| Maclaurin for \ln(1+x) | \ln(1+x)=\sum_{n=1}^{\infty}(-1)^{n-1}\frac{x^n}{n} | Converges for -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|
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|
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|
You’ve got this—use the sheet to save memory, and spend your brainpower on setup and interpretation.