AP Calculus formula sheet: what to memorise

By Ashraf Bhura, Co-Founder, STEM-Tutor · Updated October 2026

Unlike AP Physics and AP Statistics, the AP Calculus exams do not come with a formula sheet. Here is what you need to know by heart.

No sheet in the exam

The AP Calculus AB and BC exams do not provide a formula sheet, so you must know the key results yourself. Check AP Central for current exam rules, including which calculators are allowed.

Derivatives

• Power rule: d/dx(xⁿ) = nxⁿ⁻¹
• d/dx(sin x) = cos x, d/dx(cos x) = −sin x, d/dx(tan x) = sec²x
• d/dx(e^x) = e^x, d/dx(ln x) = 1/x
• Product, quotient and chain rules
• Derivatives of inverse trig functions, such as d/dx(arctan x) = 1/(1 + x²)

Integrals

• ∫xⁿ dx = xⁿ⁺¹/(n + 1) + C, for n ≠ −1
• ∫1/x dx = ln|x| + C
• ∫sin x dx = −cos x + C, ∫cos x dx = sin x + C
• u-substitution

Key theorems

• The Fundamental Theorem of Calculus, both parts
• The Mean Value Theorem and the Intermediate Value Theorem
• The Extreme Value Theorem
• L'Hôpital's rule for indeterminate limits

Applications

• Average value of a function: (1/(b − a))∫_a^bf(x) dx
• Area between curves, and volumes by discs, washers and known cross-sections
• Motion: velocity is the derivative of position; displacement is the integral of velocity

Extra for BC

• Integration by parts: ∫u dv = uv − ∫v du
• Arc length, parametric derivatives and polar area: ½∫r² dθ
• Euler's method and the logistic model
• Taylor and Maclaurin series for e^x, sin x, cos x and 1/(1 − x)
• Convergence tests: geometric, p-series, ratio, alternating series and comparison tests

How to memorise effectively

1. Make your own one-page sheet and rewrite it from memory weekly.
2. Learn why results are true where you can; it makes them stick.
3. Use them in mixed problems, not just as a list.

Worked examples using what you memorise

Example 1: Fundamental Theorem, part 2

Evaluate ∫₁⁴ √x dx.

• Antiderivative: (2/3)x^3/2.
• (2/3)(8 − 1) = 14/3.

Example 2: Fundamental Theorem, part 1

Find d/dx ∫₀^x cos(t²) dt.

• The derivative of an accumulation function is the integrand at the upper limit: cos(x²).

Example 3: Mean Value Theorem

f(x) = x² on [1, 3]. Find c where f'(c) equals the average rate of change.

• Average rate = (9 − 1)/(3 − 1) = 4.
• f'(c) = 2c = 4, so c = 2, which lies in (1, 3) as the theorem promises.

Example 4: volume by discs

Find the volume when y = √x from x = 0 to 4 is rotated about the x-axis.

• V = π∫₀⁴ (√x)² dx = π∫₀⁴ x dx = 8π.

Example 5: L'Hôpital's rule

Find lim_x→0 (e^x − 1)/x.

• It has the form 0/0. Differentiate top and bottom: lim e^x/1 = 1.

How to build your own formula sheet

1. Make one page with four sections: derivatives, integrals, theorems and applications. BC students add a fifth for series and parametric/polar results.
2. Write each result with a tiny example next to it, such as "d/dx(tan x) = sec²x, e.g. d/dx tan 3x = 3sec²3x".
3. Cover the page and rewrite it from memory once a week. Add anything you forgot to a short "weak list".
4. In the last month, rewrite only the weak list until it is empty.

Common mistakes

• Forgetting the chain rule factor, for example writing d/dx(sin 3x) = cos 3x.
• Using the Fundamental Theorem without the chain rule when the upper limit is a function of x.
• Stating the Mean Value Theorem without checking continuity and differentiability.
• Forgetting π in disc and washer volumes.

Practice questions

Try each one before opening the answer.

Frequently asked questions

Is there any formula sheet at all?

No. The AP Calculus exams do not provide one, so memorisation is essential.

Which results are tested most?

Derivative and integral rules, the Fundamental Theorem, and applications such as area, volume, motion and accumulation appear every year.

Key takeaways

• The AP Calculus exams give no formula sheet.
• Memorise derivative and integral rules with examples.
• Know both parts of the Fundamental Theorem of Calculus.
• State theorem conditions when you use them.
• Build and rewrite your own one-page sheet weekly.

Understanding beats memorising

Memorising works best when each result is tied to an idea. The Fundamental Theorem of Calculus links the two halves of the course: integration and differentiation undo each other. The Mean Value Theorem says that somewhere, the instantaneous rate equals the average rate. Disc volumes are just areas of circles added up along an axis. When you understand the idea, you can rebuild the formula even under exam pressure.

Results students forget most often

• The derivative of tan x and of the inverse trig functions.
• The integral of 1/x, which needs an absolute value: ln|x|.
• The average value formula, including the 1/(b − a) factor.
• For BC: the Lagrange error bound and the interval of convergence steps.

More questions students ask

Should I memorise the trig identities too?

Know sin²x + cos²x = 1 and the basic values at standard angles. Other identities are rarely needed in AP Calculus.

One more worked example: average value in context

The temperature in a room, in degrees Celsius, is T(t) = 18 + 2t for 0 ≤ t ≤ 5 hours. What is the average temperature over those five hours?

• Average value = (1/5)∫₀⁵ (18 + 2t) dt.
• The integral is [18t + t²] from 0 to 5 = 90 + 25 = 115.
• Average = 115/5 = 23°C. In words: over the five hours, the room was 23°C on average.

Study tip

Each week, pick five results from your sheet at random and use each one in a short question. Using a formula in context fixes it in memory far better than rereading a list.

Further questions

What if I forget a formula in the exam?

Try to rebuild it from the idea behind it, or move on and come back. Partial work, such as a correct setup, can still earn marks in free-response questions.

Is it worth memorising Riemann sum formulas?

Know how to build left, right, midpoint and trapezoidal sums from a table. That understanding matters more than a written formula.

Where to go next

Memorised results are most useful when you practise them in real questions. These guides apply them in context, with worked examples and answers.

• AP Calculus derivative rules
• AP Calculus AB review and study plan
• AP Calculus BC series

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