By This calculus formulas sheet collects the rules you use most in Calculus I and II on one searchable page: derivatives, integrals, limits, series, the big theorems and the standard applications. Type a word such as “chain”, “taylor” or “volume” into the search box to filter it, and tap TeX next to any formula to copy its LaTeX source into your notes, homework or slides.
Below the sheet you’ll find how the formulas fit together, which ones to memorize first, and a compact table of the most common derivatives and integrals. It’s free and needs no sign-up.
How to use the formula sheet
- Scroll through the six groups, or type in the Search box. It matches formula names and topic words, so “sin”, “parts”, “ftc” or “revolution” all work.
- Tap TeX to copy a formula’s LaTeX. Paste it into any LaTeX or Markdown editor that renders math, wrapped in that editor’s math delimiters (usually dollar signs).
- Clear the search box to see the full sheet again.
How the formulas fit together
The 44 formulas fall into six groups, and they build on one another:
- Limits come first. The derivative itself is a limit, ( f'(x) = lim_{h to 0} frac{f(x+h) – f(x)}{h} ), and the special limits ( frac{sin x}{x} to 1 ) and ( frac{e^x – 1}{x} to 1 ) are exactly what make the derivatives of ( sin x ) and ( e^x ) come out so cleanly.
- Derivatives split into rules for combining functions (power, product, quotient, chain) and a list of basic functions (trig, exponential, log, inverse trig).
- Integrals mirror the derivative list read backward. Also, the two techniques, u-substitution and integration by parts, are the chain rule and product rule in reverse.
- Theorems connect the two sides: the Fundamental Theorem says integrating and differentiating undo each other.
- Series and Applications reuse everything above: Taylor series are built from repeated derivatives, while arc length, volume and average value are all definite integrals.
The formulas to memorize first
If you only learn a handful by heart, make it these. Almost everything else can be derived from them.
- The power rule both ways: ( (x^n)’ = nx^{n-1} ) and ( int x^n,dx = frac{x^{n+1}}{n+1} + C ) for ( n neq -1 ).
- The chain rule: ( frac{d}{dx}f(g(x)) = f'(g(x)),g'(x) ). Most derivative mistakes on exams are chain-rule mistakes.
- The six basic derivatives: ( sin, cos, e^x, ln x, tan x ) and ( arctan x ).
- Integration by parts: ( int u,dv = uv – int v,du ).
- The Fundamental Theorem of Calculus: ( int_a^b f(x),dx = F(b) – F(a) ).
Most-used derivatives and integrals
Each integral is a derivative, read right to left. Add (+ C) to every antiderivative. To check an entry, or an unlisted function, use the derivative calculator. The integral calculator also helps with calculus formulas.
| ( f(x) ) | ( f'(x) ) | ( int f(x),dx ) |
|---|---|---|
| ( x^n ) | ( nx^{n-1} ) | ( frac{x^{n+1}}{n+1} ), ( n neq -1 ) |
| ( frac{1}{x} ) | ( -frac{1}{x^2} ) | ( lnlvert xrvert ) |
| ( e^x ) | ( e^x ) | ( e^x ) |
| ( a^x ) | ( a^x ln a ) | ( frac{a^x}{ln a} ) |
| ( ln x ) | ( frac{1}{x} ) | ( xln x – x ) |
| ( sin x ) | ( cos x ) | ( -cos x ) |
| ( cos x ) | ( -sin x ) | ( sin x ) |
| ( tan x ) | ( sec^2 x ) | ( -lnlvertcos xrvert ) |
| ( sec x ) | ( sec xtan x ) | ( lnlvertsec x + tan xrvert ) |
Two more pairs are worth knowing cold: ( (arctan x)’ = frac{1}{1+x^2} ) and ( (arcsin x)’ = frac{1}{sqrt{1-x^2}} ), so those fractions integrate straight back to ( arctan x ) and ( arcsin x ).
Worked example: combining formulas
Derivative. Differentiate ( x^2 ln x ). The product rule plus the power and log rules give
$$frac{d}{dx}left(x^2ln xright) = 2xln x + x^2cdotfrac{1}{x} = 2xln x + x$$
Integral. Evaluate ( int_0^1 frac{2x}{x^2+1},dx ). With ( u = x^2 + 1 ), ( du = 2x,dx ) (see u-substitution), the integrand becomes ( frac{1}{u} ):
$$int_0^1 frac{2x}{x^2+1},dx = Big[ln(x^2+1)Big]_0^1 = ln 2$$
By parts. For ( int xcos x,dx ), take ( u = x ), ( dv = cos x,dx ). As a result, integration by parts gives ( xsin x + cos x + C ).
Common mix-ups
- Sign of cosine. ( (cos x)’ = -sin x ), but ( int sin x,dx = -cos x ). However, the minus sign lives in a different place each time.
- The ( n = -1 ) exception. The integral power rule fails for ( frac{1}{x} ). Because, it integrates to ( lnlvert xrvert ).
- Geometric series need ( |r| < 1 ). ( sum_{n=0}^{infty} left(tfrac12right)^n = 2 ), but the same formula gives nonsense for ( r = 2 ).
Formulas for applications
| Quantity | Formula |
|---|---|
| Area between curves | \( \int_a^b [f(x) - g(x)]\,dx \) |
| Volume (disks) | \( \pi\int_a^b R(x)^2\,dx \) |
| Volume (shells) | \( 2\pi\int_a^b r(x)h(x)\,dx \) |
| Arc length | \( \int_a^b \sqrt{1 + f'(x)^2}\,dx \) |
| Average value | \( \frac{1}{b - a}\int_a^b f(x)\,dx \) |
| Work | \( \int_a^b F(x)\,dx \) |
Each has its own calculator with worked examples: area between curves, disk method, shell method, arc length, average value and work.
Useful limits and series
- \( \lim_{x \to 0} \frac{\sin x}{x} = 1 \), \( \lim_{x \to 0} \frac{e^x - 1}{x} = 1 \), \( \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n = e \)
- Geometric series: \( \sum_{n=0}^{\infty} ar^n = \frac{a}{1 - r} \) for \( |r| < 1 \)
- \( e^x = \sum \frac{x^n}{n!} \), see the Taylor series calculator
How to learn the formulas
Memorize core rules only: power, product, quotient, and chain. Learn basic derivatives and their reverse integrals. Derive other calculus formulas when needed. Practice with the calculus practice problems.
Related guides
- Power Rule for Derivatives
- Chain Rule Explained
- U-Substitution: Step-by-Step Method
- Integration by Parts
- Fundamental Theorem of Calculus
Further reading
- Differentiation Formulas (Paul’s Online Math Notes) — the basic derivative rules with many worked examples.
- Computing Indefinite Integrals (Paul’s Online Math Notes) — the integral formulas and why each one reverses a derivative rule.
FAQ
What are the most important calculus formulas?
The power rule, chain rule, and derivatives for sin, cos, e^x, and ln x are essential. Integration by parts and the Fundamental Theorem of Calculus are also key. With these calculus formulas, you can derive most other results.
How do I copy a formula in LaTeX?
Tap the TeX button on any calculus formulas card. The LaTeX source copies to your clipboard. It is ready for Overleaf or other LaTeX math editors.
Can I search for a specific formula?
Yes. The search box filters by name and topic keywords. For example, “volume” shows the disk, washer and shell methods, and “maclaurin” shows the series expansions.
Why is the integral of 1/x equal to ln|x| and not a power?
The integral power rule divides by ( n + 1 ), which is zero when ( n = -1 ). Instead, ( frac{1}{x} ) is the derivative of ( lnlvert xrvert ), so that’s its antiderivative.