Calculus Formula Cheat Sheet: Complete Reference Guide

Introduction to Calculus Formulas

Calculus is the mathematical study of continuous change, and its formulas are the foundation for understanding rates of change, accumulation, and limits. This comprehensive cheat sheet provides all essential formulas with clear explanations and practical examples.

Why This Cheat Sheet is Essential:

Consolidates all key calculus formulas in one place
Provides clear explanations with practical examples
Includes both basic and advanced formulas
Organized by topic for easy reference
Perfect for students, educators, and professionals

Whether you're studying for exams, working on engineering problems, or need a quick reference, this cheat sheet has everything you need to master calculus.

Limits & Continuity

Limits form the foundation of calculus, describing the behavior of functions as inputs approach certain values.

εδ

Limit Definition

lim_x→a f(x) = L

Meaning: As x approaches a, f(x) approaches L

Example: lim_x→2 (x² - 4)/(x - 2) = 4

Even though f(2) is undefined, the limit exists.

±∞

Infinite Limits

lim_x→a f(x) = ∞

Meaning: f(x) grows without bound as x approaches a

Example: lim_x→0 1/x² = ∞

The function approaches infinity from both sides.

→

One-Sided Limits

lim_x→a⁺ f(x) = L₊
lim_x→a⁻ f(x) = L_-

Meaning: Approach from right (+) or left (-)

Example: For f(x) = |x|/x, lim_x→0⁺ = 1, lim_x→0⁻ = -1

∞/∞

L'Hôpital's Rule

lim_x→a f(x)/g(x) = lim_x→a f'(x)/g'(x)

Condition: 0/0 or ∞/∞ form

Example: lim_x→0 sin(x)/x = lim_x→0 cos(x)/1 = 1

Continuity Conditions

A function f(x) is continuous at x = a if:

f(a) is defined
lim_x→a f(x) exists
lim_x→a f(x) = f(a)

Example: f(x) = x² is continuous everywhere

Counterexample: f(x) = 1/x is discontinuous at x = 0

Turn theory into practice with real-world problems using the surface area calculator.

Derivatives

Derivatives measure instantaneous rates of change and are fundamental to differential calculus.

Basic Differentiation Rules

Power Rule

d/dx [xⁿ] = nxⁿ⁻¹

For any real number n

Constant Rule

d/dx [c] = 0

Derivative of constant is zero

Constant Multiple

d/dx [cf(x)] = c f'(x)

Constants factor out

Sum/Difference

d/dx [f(x) ± g(x)] = f'(x) ± g'(x)

Derivative distributes over sums

Product & Quotient Rules

Product Rule

d/dx [f(x)g(x)] = f'(x)g(x) + f(x)g'(x)

First times derivative of second plus second times derivative of first

Quotient Rule

d/dx [f(x)/g(x)] = [f'(x)g(x) - f(x)g'(x)]/[g(x)]²

Low d-high minus high d-low over low squared

Chain Rule

d/dx [f(g(x))] = f'(g(x)) · g'(x)

Derivative of outside times derivative of inside

General Power Rule

d/dx [uⁿ] = nuⁿ⁻¹ · u'

Combines power rule with chain rule

Derivatives of Common Functions

Function	Derivative	Domain
eˣ	eˣ	(-∞, ∞)
aˣ	aˣ ln(a)	(-∞, ∞)
ln\|x\|	1/x	x ≠ 0
logₐ\|x\|	1/(x ln a)	x > 0
sin x	cos x	(-∞, ∞)
cos x	-sin x	(-∞, ∞)
tan x	sec² x	x ≠ π/2 + nπ
sin⁻¹ x	1/√(1-x²)	\|x\| < 1
tan⁻¹ x	1/(1+x²)	(-∞, ∞)

Derivative Calculator

Enter function f(x)

Enter a function and click "Calculate"

If you're ready to practice, apply concepts in real scenarios with the surface area calculator.

Applications of Derivatives

Derivatives have numerous practical applications in optimization, motion, and curve analysis.

↕️

Tangent Line

y - y₁ = f'(x₁)(x - x₁)

Equation: Point-slope form using derivative as slope

Example: For f(x) = x² at (2,4): y - 4 = 4(x - 2)

📈

Increasing/Decreasing

f'(x) > 0 ⇒ f increasing
f'(x) < 0 ⇒ f decreasing

Test: Sign of first derivative

⛰️

First Derivative Test

f'(c) = 0 and sign change ⇒ local extremum

For maxima/minima: + to - ⇒ max, - to + ⇒ min

📉

Second Derivative Test

f''(c) > 0 ⇒ local minimum
f''(c) < 0 ⇒ local maximum

Condition: f'(c) = 0

Optimization Problems

Steps to solve optimization problems:

Identify quantity to optimize
Write equation relating variables
Express quantity as function of one variable
Find critical points using f'(x) = 0
Test endpoints and critical points

Example: Maximize area of rectangle with fixed perimeter P

A = x(P/2 - x), dA/dx = P/2 - 2x = 0 ⇒ x = P/4 (square maximizes area)

Integrals

Integration is the reverse process of differentiation and is used to find areas, volumes, and accumulations.

Basic Integration Rules

Power Rule

∫xⁿ dx = xⁿ⁺¹/(n+1) + C, n ≠ -1

Reverse of power rule for derivatives

Constant Multiple

∫c f(x) dx = c ∫f(x) dx

Constants factor out

Sum/Difference

∫[f(x) ± g(x)] dx = ∫f(x) dx ± ∫g(x) dx

Integral distributes over sums

∫1/x dx

∫1/x dx = ln|x| + C

Special case when n = -1

Integration Techniques

Substitution

∫f(g(x))g'(x) dx = ∫f(u) du

Let u = g(x), then du = g'(x) dx

Integration by Parts

∫u dv = uv - ∫v du

Based on product rule for derivatives

Trigonometric Integrals

∫sin x dx = -cos x + C
∫cos x dx = sin x + C

Basic trigonometric integrals

Definite Integrals

Fundamental Theorem

∫_a^b f(x) dx = F(b) - F(a)

Where F'(x) = f(x)

Properties

∫_a^b f(x) dx = -∫_b^a f(x) dx

Reversing limits changes sign

Additivity

∫_a^b f(x) dx + ∫_b^c f(x) dx = ∫_a^c f(x) dx

Integrals over adjacent intervals

Integral Calculator

Enter function f(x)

Lower limit (optional)

Upper limit (optional)

Enter a function and limits (optional) and click "Calculate"

Want to evaluate your knowledge? Solve real-life problems using the surface area calculator.

Applications of Integrals

Integrals are used to compute areas, volumes, work, and many other physical quantities.

📏

Area Between Curves

A = ∫_a^b [f(x) - g(x)] dx

Where: f(x) ≥ g(x) on [a, b]

Example: Area between y = x² and y = x from 0 to 1

A = ∫₀¹ (x - x²) dx = 1/6

📦

Volume by Slicing

V = ∫_a^b A(x) dx

Where: A(x) is cross-sectional area

🔄

Disk Method

V = π ∫_a^b [R(x)]² dx

Rotation: About x-axis, R(x) = f(x)

⚙️

Work

W = ∫_a^b F(x) dx

Where: F(x) is variable force

Example: Work to stretch spring: W = ∫₀^x kx dx = ½kx²

Arc Length

The length of a curve y = f(x) from x = a to x = b:

L = ∫_a^b √[1 + (f'(x))²] dx

Example: Arc length of y = x³/² from 0 to 1

f'(x) = (3/2)x¹/², L = ∫₀¹ √[1 + (9/4)x] dx ≈ 1.44

Infinite Series

Series represent the sum of infinitely many terms and are crucial for approximations and analysis.

∑

Geometric Series

∑_n=0^∞ arⁿ = a/(1-r), |r| < 1

Converges: When |r| < 1

Example: ∑_n=0^∞ (1/2)ⁿ = 1/(1-½) = 2

p

p-Series

∑_n=1^∞ 1/nᵖ

Converges: When p > 1

Example: ∑ 1/n² converges (p = 2 > 1)

∑ 1/n diverges (p = 1 ≤ 1)

!n

Taylor Series

f(x) = ∑_n=0^∞ f⁽ⁿ⁾(a)(x-a)ⁿ/n!

About: x = a (Maclaurin if a = 0)

eˣ

Common Series

eˣ = ∑ xⁿ/n!
sin x = ∑ (-1)ⁿ x²ⁿ⁺¹/(2n+1)!
cos x = ∑ (-1)ⁿ x²ⁿ/(2n)!

All: Converge for all real x

Convergence Tests

Test	Condition	Conclusion
Divergence Test	lim aₙ ≠ 0	Diverges
Integral Test	f positive, continuous, decreasing	∑aₙ and ∫f(x) dx same behavior
Comparison Test	0 ≤ aₙ ≤ bₙ	If ∑bₙ converges, ∑aₙ converges
Ratio Test	lim \|aₙ₊₁/aₙ\| = L	L < 1: converges, L > 1: diverges
Root Test	lim \|aₙ\|¹/ⁿ = L	L < 1: converges, L > 1: diverges

To check your understanding, work through practical examples with the surface area calculator.

Multivariable Calculus

Extending calculus to functions of several variables for modeling real-world phenomena.

∂

Partial Derivatives

∂f/∂x = lim_h→0 [f(x+h,y)-f(x,y)]/h

Meaning: Derivative with respect to x, holding y constant

∇

Gradient

∇f = (∂f/∂x, ∂f/∂y, ∂f/∂z)

Direction: Points in direction of steepest ascent

∭

Multiple Integrals

∬_R f(x,y) dA
∭_D f(x,y,z) dV

Volume: ∭_D dV gives volume of region D

∮

Line Integrals

∫_C f ds = ∫_a^b f(r(t))|r'(t)| dt

Parametric: C given by r(t), a ≤ t ≤ b

Chain Rule for Multiple Variables

If z = f(x,y) where x = g(t) and y = h(t):

dz/dt = (∂f/∂x)(dx/dt) + (∂f/∂y)(dy/dt)

Example: If z = x²y, x = t, y = t², then

dz/dt = (2xy)(1) + (x²)(2t) = 2t³ + 2t³ = 4t³

Differential Equations

Equations involving derivatives that model rates of change in various fields.

y'

Separable Equations

dy/dx = g(x)h(y)
∫dy/h(y) = ∫g(x) dx

Solution: Separate variables and integrate

λ

First Order Linear

dy/dx + P(x)y = Q(x)
y = e⁻∫ᴾ[∫Qe∫ᴾ dx + C]

Integrating factor: μ = e∫ᴾ⁽ˣ⁾ ᵈˣ

ω

Second Order Constant

ay'' + by' + cy = 0
Characteristic: ar² + br + c = 0

Solution: Based on roots r₁, r₂

→

Growth & Decay

dP/dt = kP
P(t) = P₀eᵏᵗ

k > 0: Growth, k < 0: Decay

Example: Radioactive decay: A(t) = A₀e⁻ᵏᵗ

Solution Types for ay'' + by' + cy = 0

Roots	General Solution
r₁ ≠ r₂ (real)	y = C₁eʳ¹ˣ + C₂eʳ²ˣ
r₁ = r₂ = r	y = (C₁ + C₂x)eʳˣ
r = α ± iβ	y = eᵅˣ(C₁cosβx + C₂sinβx)

If you want to test your skills, explore real-world applications using the surface area calculator.

Quick Reference Guide

Essential formulas at a glance for quick problem-solving.

Derivative Shortcuts

d/dx sin x = cos x

Sine

d/dx cos x = -sin x

Cosine

d/dx tan x = sec² x

Tangent

d/dx eˣ = eˣ

Exponential

d/dx ln x = 1/x

Natural Log

d/dx sin⁻¹ x = 1/√(1-x²)

Arcsin

Integration Shortcuts

∫sin x dx = -cos x + C