Calculus formula sheet: derivatives, integrals, limits and trig

The Calculus 1 formulas in one place, ready to print on one double-sided A4 sheet. The same formulas as the full tables, arranged by topic.

Table of derivatives: Differentiation rules

Expression	Result
$(c f)^{'}$	$c f^{'}$
$(f \pm g)^{'}$	$f^{'} \pm g^{'}$
$(f g)^{'}$	$f^{'} g + f g^{'}$
$(\frac{f}{g})^{'}$	$\frac{f ^{'} g - f g ^{'}}{g ^{2}}$
$(f (g (x)))^{'}$	$f^{'} (g (x)) \cdot g^{'} (x)$

Table of derivatives: Powers, exponentials and logarithms

Expression	Result
$c$	$0$
$x^{n}$	$n x^{n - 1}$
$x$	$\frac{1}{2 x}$
$e^{x}$	$e^{x}$
$a^{x}$	$a^{x} ln a$
$ln x$	$\frac{1}{x}$
$lo g_{a} x$	$\frac{1}{x ln a}$

Table of derivatives: Trigonometric, inverse and hyperbolic

Expression	Result
$sin x$	$cos x$
$cos x$	$- sin x$
$tan x$	$\frac{1}{cos ^{2} x}$
$cot x$	$- \frac{1}{sin ^{2} x}$
$arcsin x$	$\frac{1}{1 - x ^{2}}$
$arccos x$	$- \frac{1}{1 - x ^{2}}$
$arctan x$	$\frac{1}{1 + x ^{2}}$
$sinh x$	$cosh x$
$cosh x$	$sinh x$

Table of integrals: Methods: by parts, substitution, linear argument

Expression	Result
$\int u d v$	$u v - \int v d u$
$\int f (g (x)) g^{'} (x) d x$	$\int f (u) d u, u = g (x)$
$\int f (a x + b) d x$	$\frac{1}{a} F (a x + b) + C$

Table of integrals: Powers, exponentials and logarithms

Expression	Result
$\int x^{n} d x$	$\frac{x ^{n + 1}}{n + 1} + C, n \neq = - 1$
$\int \frac{1}{x} d x$	$ln ∣ x ∣ + C$
$\int e^{x} d x$	$e^{x} + C$
$\int a^{x} d x$	$\frac{a ^{x}}{ln a} + C$
$\int ln x d x$	$x ln x - x + C$

Table of integrals: Trigonometric

Expression	Result
$\int sin x d x$	$- cos x + C$
$\int cos x d x$	$sin x + C$
$\int tan x d x$	$- ln ∣ cos x ∣ + C$
$\int \frac{1}{cos ^{2} x} d x$	$tan x + C$
$\int \frac{1}{sin ^{2} x} d x$	$- cot x + C$

Table of integrals: Integrals giving arctan, arcsin and logarithms

Expression	Result
$\int \frac{1}{1 + x ^{2}} d x$	$arctan x + C$
$\int \frac{1}{a ^{2} + x ^{2}} d x$	$\frac{1}{a} arctan \frac{x}{a} + C$
$\int \frac{1}{1 - x ^{2}} d x$	$arcsin x + C$
$\int \frac{1}{a ^{2} - x ^{2}} d x$	$arcsin \frac{x}{a} + C$
$\int \frac{1}{x ^{2} - a ^{2}} d x$	$\frac{1}{2 a} ln \frac{x - a}{x + a} + C$
$\int \frac{1}{x ^{2} + a} d x$	$ln x + x^{2} + a + C$

Special limits: 0/0 limits near zero

Expression	Result
$x \to 0 lim \frac{sin x}{x}$	$1$
$x \to 0 lim \frac{tan x}{x}$	$1$
$x \to 0 lim \frac{1 - cos x}{x ^{2}}$	$\frac{1}{2}$
$x \to 0 lim \frac{e ^{x} - 1}{x}$	$1$
$x \to 0 lim \frac{a ^{x} - 1}{x}$	$ln a$
$x \to 0 lim \frac{ln ( 1 + x )}{x}$	$1$

Special limits: The limits of e

Expression	Result
$n \to \infty lim (1 + \frac{1}{n})^{n}$	$e$
$x \to \infty lim (1 + \frac{a}{x})^{x}$	$e^{a}$
$x \to 0 lim (1 + x)^{1/ x}$	$e$

Trigonometric identities: Table of values

$α$	$sin α$	$cos α$	$tan α$
$0$	$0$	$1$	$0$
$\frac{π}{6} = 3 0^{\circ}$	$\frac{1}{2}$	$\frac{3}{2}$	$\frac{3}{3}$
$\frac{π}{4} = 4 5^{\circ}$	$\frac{2}{2}$	$\frac{2}{2}$	$1$
$\frac{π}{3} = 6 0^{\circ}$	$\frac{3}{2}$	$\frac{1}{2}$	$3$
$\frac{π}{2} = 9 0^{\circ}$	$1$	$0$	undefined

Trigonometric identities: Basic identities

Expression	Result
$sin^{2} α + cos^{2} α$	$1$
$1 + tan^{2} α$	$\frac{1}{cos ^{2} α}$
$tan α$	$\frac{sin α}{cos α}$
$sin (- α)$	$- sin α$
$cos (- α)$	$cos α$

Trigonometric identities: Double angle and power reduction

Expression	Result
$sin 2 α$	$2 sin α cos α$
$cos 2 α$	$cos^{2} α - sin^{2} α = 2 cos^{2} α - 1 = 1 - 2 sin^{2} α$
$tan 2 α$	$\frac{2 tan α}{1 - tan ^{2} α}$
$sin^{2} α$	$\frac{1 - cos 2 α}{2}$
$cos^{2} α$	$\frac{1 + cos 2 α}{2}$

Notes

Before an exam, check your course syllabus for whether a formula sheet is allowed and which. This one covers what comes up in most first calculus courses.

Each topic has a full page with more rows and explanations: the derivatives table, the integrals table, special limits and trig identities.