AP Calculus AB/BC Formula Sheet
What You Need to Know
The AP Calculus AB/BC formula sheet (the one College Board provides) is a minimal safety net, not a full list of everything you need. Your job is to (1) know exactly what’s on it, (2) know how to combine it with rules you must memorize (especially chain rule, power rule, u-sub, etc.), and (3) avoid sign/interval traps.
Two big ideas:
- The sheet gives you core derivative/integral pairs, FTC statements, integration by parts, and (BC) standard Maclaurin series.
- You still must supply the structure: algebra, trig identities, chain rule, and correct limits/intervals.
Critical reminder: The sheet’s formulas are usually for x, but almost every exam problem is for u(x). That means chain rule (for derivatives) and u-sub (for integrals) are the hidden steps.
Step-by-Step Breakdown
A. How to use the formula sheet efficiently (on any problem)
- Identify the task type
- Derivative? Integral? Accumulation/area? Series approximation?
- Match the pattern to a formula on the sheet
- Example: see \sec^2(x) → think \tan(x).
- See \int_a^x f(t)\,dt → think FTC + chain rule.
- Check for an “inside function” u(x)
- If differentiating: apply chain rule to the sheet formula.
- If integrating: look for a derivative factor and do u-sub.
- Do a quick sign + constant check
- Indefinite integrals need +C.
- Trig pairs have common sign errors (especially \int \sin(x)\,dx).
- If it’s a series question (BC)
- Start from a known Maclaurin series on the sheet.
- Substitute/scale to match the target.
- State interval of convergence after substitution.
B. Micro-worked examples (method in action)
FTC + chain rule
- Given g(x)=\int_2^{x^3} \sqrt{1+t^2}\,dt
- Use FTC: \frac{d}{dx}\int_2^{u(x)} f(t)\,dt = f(u(x))\,u'(x)
- So g'(x)=\sqrt{1+(x^3)^2}\cdot 3x^2 = 3x^2\sqrt{1+x^6}
Integration by parts (on the sheet)
- Compute \int x e^x\,dx
- Use \int u\,dv = uv-\int v\,du
- Pick u=x, dv=e^x dx → du=dx, v=e^x
- Result: x e^x-\int e^x\,dx = x e^x - e^x + C = e^x(x-1)+C
Series substitution (BC)
- Want Maclaurin for \sin(3x)
- From sheet: \sin(x)=\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n+1}}{(2n+1)!}
- Substitute x\mapsto 3x:
\sin(3x)=\sum_{n=0}^{\infty}(-1)^n\frac{(3x)^{2n+1}}{(2n+1)!}
Key Formulas, Rules & Facts
A. Limits & derivative definition (know what it means)
| Formula | When to use | Notes |
|---|---|---|
| f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h} | Definition of derivative | Use when asked “from first principles,” or to justify differentiability. |
B. Core derivative rules (NOT fully given on the sheet—memorize)
| Rule | Formula | Notes |
|---|---|---|
| Power rule | \frac{d}{dx}x^n = n x^{n-1} | Works for any real n where defined. |
| Constant multiple | \frac{d}{dx}[c\,f(x)] = c\,f'(x) | Factor constants out. |
| Sum/difference | \frac{d}{dx}[f(x)\pm g(x)] = f'(x)\pm g'(x) | Differentiate term-by-term. |
| Product rule | (fg)'=f'g+fg' | Don’t multiply out if it’s messy. |
| Quotient rule | \left(\frac{f}{g}\right)'=\frac{f'g-fg'}{g^2} | Often easier to rewrite as f\cdot g^{-1}. |
| Chain rule | \frac{d}{dx}f(u(x))=f'(u(x))\,u'(x) | The #1 “hidden” step with the formula sheet. |
C. Derivatives of common functions (these are the classic “table” items)
| Function | Derivative | Notes |
|---|---|---|
| \sin(x) | \cos(x) | |
| \cos(x) | -\sin(x) | Sign trap. |
| \tan(x) | \sec^2(x) | |
| \cot(x) | -\csc^2(x) | |
| \sec(x) | \sec(x)\tan(x) | |
| \csc(x) | -\csc(x)\cot(x) | |
| \ln(x) | \frac{1}{x} | Domain x>0. |
| e^x | e^x | |
| a^x | a^x\ln(a) | Requires a>0, a\neq 1. |
D. Derivatives of inverse trig (common MCQ/FRQ targets)
| Function | Derivative | Notes |
|---|---|---|
| \arcsin(x) | \frac{1}{\sqrt{1-x^2}} | Domain typically |x| |
| \arccos(x) | -\frac{1}{\sqrt{1-x^2}} | Same denominator, negative sign. |
| \arctan(x) | \frac{1}{1+x^2} | Defined for all real x. |
E. Core integration facts (FTC + antiderivatives)
| Fact | Formula | Notes |
|---|---|---|
| FTC Part 1 | \frac{d}{dx}\int_a^x f(t)\,dt=f(x) | If upper limit is u(x), multiply by u'(x). |
| FTC Part 2 | \int_a^b f(x)\,dx=F(b)-F(a) | Requires F'(x)=f(x). |
| Average value | f_{\text{avg}}=\frac{1}{b-a}\int_a^b f(x)\,dx | Common FRQ for “average value on [a,b].” |
F. Indefinite integrals of trig “pairs” (watch signs)
| Integrand | Antiderivative | Notes |
|---|---|---|
| \sin(x) | -\cos(x)+C | Sign trap. |
| \cos(x) | \sin(x)+C | |
| \sec^2(x) | \tan(x)+C | |
| \csc^2(x) | -\cot(x)+C | Negative. |
| \sec(x)\tan(x) | \sec(x)+C | |
| \csc(x)\cot(x) | -\csc(x)+C | Negative. |
G. Integration techniques you must know (some are on the sheet, some aren’t)
| Technique | Formula / trigger | Notes |
|---|---|---|
| u-sub | If you see f'(x)\,g(f(x)) | Reverse chain rule. |
| Integration by parts | \int u\,dv = uv-\int v\,du | Choose u to simplify after differentiating. |
| Basic exponential/log integrals | \int \frac{1}{x}\,dx=\ln|x|+C, \int e^x dx=e^x+C | Absolute value for \ln|x| is essential. |
H. BC-only: standard Maclaurin series (know them cold)
| Function | Maclaurin series | Interval of convergence |
|---|---|---|
| \frac{1}{1-x} | \sum_{n=0}^{\infty} x^n = 1+x+x^2+\cdots | |x| |
| e^x | \sum_{n=0}^{\infty}\frac{x^n}{n!} | All real x |
| \sin(x) | \sum_{n=0}^{\infty}(-1)^n\frac{x^{2n+1}}{(2n+1)!} | All real x |
| \cos(x) | \sum_{n=0}^{\infty}(-1)^n\frac{x^{2n}}{(2n)!} | All real x |
| \ln(1+x) | \sum_{n=1}^{\infty}(-1)^{n-1}\frac{x^n}{n} | -1 |
| \arctan(x) | \sum_{n=0}^{\infty}(-1)^n\frac{x^{2n+1}}{2n+1} | |x|\le 1 |
Series substitution rule: If a series converges for |x|
Examples & Applications
1. Recognize a “reverse chain rule” integral
Compute \int \frac{2x}{1+x^2}\,dx.
- Spot inside: 1+x^2 with derivative 2x.
- Let u=1+x^2, du=2x\,dx.
- Integral becomes \int \frac{1}{u}\,du = \ln|u|+C = \ln(1+x^2)+C.
2. Use the trig integral pairs correctly
Compute \int \csc(x)\cot(x)\,dx.
- From sheet: \int \csc(x)\cot(x)\,dx = -\csc(x)+C.
- Fast check: derivative of \csc(x) is -\csc(x)\cot(x), so the negative is required.
3. FTC Part 1 with a nontrivial upper limit
Let h(x)=\int_{-1}^{\cos(x)} \frac{1}{1+t^2}\,dt. Find h'(x).
- Use FTC + chain: h'(x)=\frac{1}{1+(\cos(x))^2}\cdot \frac{d}{dx}[\cos(x)]
- h'(x)=\frac{1}{1+\cos^2(x)}\cdot (-\sin(x))= -\frac{\sin(x)}{1+\cos^2(x)}
4. Build a new series from a known one (BC)
Find the first three nonzero terms of the Maclaurin series for \ln(1-2x).
- Start: \ln(1+x)=\sum_{n=1}^{\infty}(-1)^{n-1}\frac{x^n}{n}
- Substitute x\mapsto -2x:
\ln(1-2x)=\sum_{n=1}^{\infty}(-1)^{n-1}\frac{(-2x)^n}{n} - First terms:
- n=1: (-1)^0\frac{-2x}{1}=-2x
- n=2: (-1)^1\frac{4x^2}{2}=-2x^2
- n=3: (-1)^2\frac{-8x^3}{3}=-\frac{8}{3}x^3
- So \ln(1-2x)= -2x-2x^2-\frac{8}{3}x^3+\cdots
- Interval: need -1< -2x \le 1 → -\frac{1}{2}\le x < \frac{1}{2}.
Common Mistakes & Traps
Forgetting chain rule with sheet derivatives
- Wrong: differentiating \sin(5x) as \cos(5x).
- Right: \frac{d}{dx}\sin(5x)=\cos(5x)\cdot 5.
- Fix: circle the “inside” u(x) and multiply by u'(x).
Messing up the signs on trig antiderivatives
- Common wrong ones: \int \sin(x)dx=\cos(x)+C, \int \csc^2(x)dx=\cot(x)+C.
- Fix: memorize the “negative” ones: \sin\to -\cos, \csc^2\to -\cot, \csc\cot\to -\csc.
Dropping +C on indefinite integrals
- On FRQs, missing +C can cost easy points.
- Fix: if there are no bounds, there must be +C.
Confusing \ln(x) vs \ln|x| in integrals
- Correct: \int \frac{1}{x}\,dx=\ln|x|+C.
- Fix: absolute value is required because \frac{d}{dx}\ln|x|=\frac{1}{x} for x\ne 0.
Inverse trig derivative mix-ups
- Classic trap: swapping denominators for \arcsin(x) and \arctan(x).
- Fix:
- \arcsin,\arccos → \sqrt{1-x^2}
- \arctan → 1+x^2
Using a Maclaurin series but not updating the interval after substitution
- Example: \ln(1+x) interval is -1
Treating the geometric series as valid at endpoints
- For \sum x^n, you must have |x|
Misreading FTC Part 1 when the variable is in the lower limit
- If G(x)=\int_x^a f(t)\,dt, then G'(x)=-f(x).
- Fix: flip limits to match \int_a^x and add a negative.
Memory Aids & Quick Tricks
| Trick / mnemonic | Helps you remember | When to use |
|---|---|---|
| “Sine → Cosine, Cosine → Negative Sine” | \frac{d}{dx}\sin(x)=\cos(x) and \frac{d}{dx}\cos(x)=-\sin(x) | Fast derivative recall |
| “A-S-A vs A-T-A” | \arcsin/\arccos have \sqrt{1-x^2}; \arctan has 1+x^2 | Inverse trig derivatives |
| “LIATE” (or ILATE) | Choose u in parts: Log, Inverse trig, Algebraic, Trig, Exponential | Integration by parts selection |
| “Geometric = ratio inside absolute less than 1” | For \sum r^n you need |r| | |
| “Upper limit = plug; chain rule multiplier” | \frac{d}{dx}\int_a^{u(x)} f(t)dt = f(u(x))u'(x) | FTC Part 1 problems |
Quick Review Checklist
- You can instantly locate and use the sheet’s trig derivative/integral pairs (with correct signs).
- You automatically apply chain rule when the input isn’t just x.
- You know FTC Part 1 and 2 and handle variable limits (including the negative when x is the lower limit).
- You never forget +C for indefinite integrals.
- You have inverse trig derivatives nailed: \arcsin/\arccos → \sqrt{1-x^2}, \arctan → 1+x^2.
- For BC series, you can write the 6 standard Maclaurin series and adjust intervals after substitution.
- You can use integration by parts quickly and choose u so the integral simplifies.
You’ve got this—use the sheet as a trigger, and let your rules (chain rule, u-sub, FTC) do the real work.