Calculus Cheat Sheet · Print-optimized · 3-column layout

Calculus Cheat Sheet

Limits · Derivatives · Integrals · Series — the formulas, rules, and theorems you actually need

Limits — Definitions

$\lim_{x \to a} f(x) = L$ iff $f(x) \to L$ as $x \to a$ from both sides.

One-sided: $\lim_{x \to a^-}$ (left), $\lim_{x \to a^+}$ (right). Two-sided limit exists iff both one-sided limits exist and are equal.

At infinity: $\lim_{x \to \pm\infty} f(x) = L$ iff $f(x) \to L$ as $x$ grows without bound.

Limit Laws

Assume $\lim f$ and $\lim g$ exist. Then:

Sum$\lim (f+g) = \lim f + \lim g$

Difference$\lim (f-g) = \lim f - \lim g$

Constant multiple$\lim (c\,f) = c \lim f$

Product$\lim (fg) = \lim f \cdot \lim g$

Quotient$\lim (f/g) = \frac{\lim f}{\lim g}$, if $\lim g \ne 0$

Power$\lim [f(x)]^n = [\lim f(x)]^n$

Root$\lim \sqrt[n]{f} = \sqrt[n]{\lim f}$

Squeeze: if $g \le f \le h$ and $\lim g = \lim h = L$, then $\lim f = L$.

Indeterminate Forms

Direct substitution gives no info. Need a tool:

$\frac{0}{0}, \ \frac{\infty}{\infty}, \ 0\cdot\infty, \ \infty-\infty, \ 0^0, \ 1^\infty, \ \infty^0$

For $\frac{0}{0}$ rationals: factor and cancel. For $\frac{\infty}{\infty}$ rationals: leading-term extraction. Else: L'Hôpital.

Fraction Arithmetic

The two recipes that come up constantly when $f$ is a rational function (e.g. $f(x) = c/x$).

Common denominator$\dfrac{a}{b} \pm \dfrac{c}{d} = \dfrac{ad \pm cb}{bd}$

Multiplication$\dfrac{a}{b} \cdot \dfrac{c}{d} = \dfrac{ac}{bd}$

Division$\dfrac{a/b}{c/d} = \dfrac{a}{b} \cdot \dfrac{d}{c} = \dfrac{ad}{bc}$

For the common-denominator recipe, distribute the $\pm$ into both pieces of the second numerator. Easy to drop a sign on the $cb$ term. Watch the negatives.

Distribute the minus sign before combining: $\dfrac{a}{b} - \dfrac{c}{d} = \dfrac{a}{b} + \dfrac{-c}{d}$, then apply the recipe with the flipped numerator. Cleaner than trying to remember which signs go where all at once.

Important Special Limits

Sinc$\lim_{x \to 0} \frac{\sin x}{x} = 1$

$e$$\lim_{x \to 0} \frac{e^x - 1}{x} = 1$

$e$$\lim_{x \to 0} (1+x)^{1/x} = e$

Log$\lim_{x \to 0^+} x \ln x = 0$

$1^\infty$$\lim (1 + \frac{a}{x})^x = e^a$

Continuity at $a$

(1) $f(a)$ defined (2) $\lim_{x\to a} f(x)$ exists (3) $f(a) = \lim_{x\to a} f(x)$

All polynomials, $\sin, \cos, e^x, \ln$ on their domains are continuous everywhere on their domain.

Discontinuity Types

Removable (hole): limit exists, function value doesn't match. Fix: redefine $f(a)$.

Jump: one-sided limits exist but disagree.

Infinite (vertical asymptote): one or both one-sided limits $\to \pm\infty$.

Slope and Equations of Lines

Two points determine a line. To write its equation: get the slope, then pick a point.

Slope (from two points)

For $(x_1, y_1)$ and $(x_2, y_2)$:

$$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\Delta y}{\Delta x}$$

Rise over run. Order must match top-to-bottom and left-to-right.

Point-slope form

$$y - y_1 = m(x - x_1)$$

Use this when you know $m$ and any point $(x_1, y_1)$ on the line.

Slope-intercept form

$$y = mx + b$$

$m$ = slope, $b$ = $y$-intercept (the point $(0,b)$).

Special slopes

Horizontal$m = 0$ → $y = c$

Verticalundefined → $x = c$

Parallel lines$m_1 = m_2$

Perpendicular$m_1 \cdot m_2 = -1$

Difference Quotient (Secant Slope)

Slope of a line through two points on a curve — the average rate of change of $f$ between those points:

$$m_{sec} = \frac{f(a+h) - f(a)}{h} = \frac{\Delta y}{\Delta x} \quad \text{at } x = a$$

$$m_{sec} = \frac{f(x+h) - f(x)}{h} = \frac{\Delta y}{\Delta x} \quad \text{at any point } x$$

Three names for the same thing: difference quotient, slope of secant line, average rate of change.

Definition of Derivative (Tangent Slope)

Take the limit of the difference quotient as $h \to 0$ — the secant line "rotates" to become the tangent line:

$$f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} = \lim_{x \to a} \frac{f(x) - f(a)}{x-a}$$

Three names for the same thing: limit of the difference quotient, slope of the tangent line, instantaneous rate of change.

Tangent line at $a$: $\ y - f(a) = f'(a)(x-a)$

Notation

$f'(x) = \dfrac{df}{dx} = \dfrac{d}{dx}[f(x)] = y' = \dot y = D_x f$

Higher: $f''(x), \ f^{(n)}(x)$

Differentiation Rules

Constant$\frac{d}{dx}(c) = 0$

Power$\frac{d}{dx}(x^n) = nx^{n-1}$

Constant mult.$(cf)' = c\,f'$

Sum/Diff$(f \pm g)' = f' \pm g'$

Product$(fg)' = f'g + fg'$

Quotient$\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}$

Chain$\frac{d}{dx} f(g(x)) = f'(g(x))\,g'(x)$

Common Derivatives

$\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$

$e^x$$e^x$

$a^x$$a^x \ln a$

$\ln x$$1/x$

$\log_a x$$1/(x \ln a)$

$\arcsin x$$1/\sqrt{1-x^2}$

$\arctan x$$1/(1+x^2)$

Inverse function

$(f^{-1})'(x) = \dfrac{1}{f'(f^{-1}(x))}$

Logarithmic differentiation

For $y = [f(x)]^{g(x)}$: take $\ln$ of both sides, differentiate implicitly.

Definition of Integral

$$\int_a^b f(x)\,dx = \lim_{n\to\infty} \sum_{i=1}^n f(x_i^*)\,\Delta x$$

(Riemann sum. $\Delta x = (b-a)/n$, $x_i^*$ in $[x_{i-1}, x_i]$.)

Fundamental Theorem of Calculus

Part 1: $\dfrac{d}{dx}\!\int_a^x f(t)\,dt = f(x)$

Part 2: $\int_a^b f(x)\,dx = F(b) - F(a)$ where $F' = f$

Integration Rules

Constant mult.$\int c\,f\,dx = c\int f\,dx$

Sum/Diff$\int (f \pm g) = \int f \pm \int g$

Power$\int x^n\,dx = \frac{x^{n+1}}{n+1} + C$ (if $n\ne -1$)

$1/x$$\int \frac{dx}{x} = \ln|x| + C$

$e^x$$\int e^x\,dx = e^x + C$

$a^x$$\int a^x\,dx = \frac{a^x}{\ln a} + C$

Common Integrals

$\int \sin x\,dx$$-\cos x + C$

$\int \cos x\,dx$$\sin x + C$

$\int \sec^2 x\,dx$$\tan x + C$

$\int \csc^2 x\,dx$$-\cot x + C$

$\int \sec x \tan x\,dx$$\sec x + C$

$\int \csc x \cot x\,dx$$-\csc x + C$

$\int \tan x\,dx$$\ln|\sec x| + C$

$\int \cot x\,dx$$\ln|\sin x| + C$

$\int \sec x\,dx$$\ln|\sec x + \tan x| + C$

$\int \frac{dx}{\sqrt{a^2 - x^2}}$$\arcsin(x/a) + C$

$\int \frac{dx}{a^2 + x^2}$$\frac{1}{a}\arctan(x/a) + C$

$\int \frac{dx}{x\sqrt{x^2 - a^2}}$$\frac{1}{a}\operatorname{arcsec}|x/a| + C$

Integration Techniques

u-substitution: $\int f(g(x))\,g'(x)\,dx = \int f(u)\,du$ where $u = g(x)$.

Integration by parts: $\int u\,dv = uv - \int v\,du$. (LIATE for choosing $u$: Logs, Inverse trig, Algebraic, Trig, Exp.)

Partial fractions: for $\dfrac{P(x)}{Q(x)}$ where $\deg P < \deg Q$ and $Q$ factors into linear/irreducible quadratics.

Trig substitution:

$\sqrt{a^2 - x^2}$: let $x = a\sin\theta$
$\sqrt{a^2 + x^2}$: let $x = a\tan\theta$
$\sqrt{x^2 - a^2}$: let $x = a\sec\theta$

Key Theorems

Mean Value Theorem: if $f$ continuous on $[a,b]$ and differentiable on $(a,b)$, then $\exists\, c \in (a,b)$ with $f'(c) = \frac{f(b)-f(a)}{b-a}$.

Intermediate Value Theorem: continuous $f$ on $[a,b]$ takes every value between $f(a)$ and $f(b)$.

Extreme Value Theorem: continuous $f$ on $[a,b]$ attains its max and min.

Rolle's Theorem: MVT with $f(a) = f(b)$: $\exists\, c$ with $f'(c) = 0$.

L'Hôpital's Rule: if $\frac{f}{g} \to \frac{0}{0}$ or $\frac{\infty}{\infty}$, then $\lim \frac{f}{g} = \lim \frac{f'}{g'}$ (when the latter limit exists).

Taylor series: $f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!}(x-a)^n$

Maclaurin series = Taylor with $a=0$. Common: $e^x = \sum x^n/n!$, $\sin x = \sum (-1)^n x^{2n+1}/(2n+1)!$, $\cos x = \sum (-1)^n x^{2n}/(2n)!$, $\frac{1}{1-x} = \sum x^n$ for $|x| < 1$.

Binomial Theorem

$$(a+b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k, \quad \binom{n}{k} = \frac{n!}{k!(n-k)!}$$

Trinomial: $a^2 + 2ab + b^2 = (a+b)^2$. $a^2 - b^2 = (a-b)(a+b)$ (difference of squares).