Calculus Reference

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Essential Formulas & Examples

Essential calculus formulas for derivatives, integrals, limits, and series

🔢Important Calculus Constants

Euler's Number

2.71828...

Natural exponential and logarithms

3.14159...

Trigonometric functions and areas

Golden Ratio

1.61803...

Special sequences and limits

Euler-Mascheroni

0.57721...

Harmonic series and integrals

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Basic Derivatives

Constant Rule

d/dx[c] = 0

Derivative of constant is zero

Power Rule

d/dx[xⁿ] = nxⁿ⁻¹

Bring down exponent, subtract 1 from power

Constant Multiple

d/dx[cf(x)] = c·f'(x)

Factor out constants

Sum Rule

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

Derivative of sum equals sum of derivatives

Difference Rule

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

Derivative of difference equals difference of derivatives

Natural Log

d/dx[ln(x)] = 1/x

Derivative of natural logarithm

Exponential

d/dx[eˣ] = eˣ

Exponential function is its own derivative

General Exponential

d/dx[aˣ] = aˣ·ln(a)

Exponential with arbitrary base

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Advanced Derivative 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, all over low squared

Chain Rule

d/dx[f(g(x))] = f'(g(x))·g'(x)

Derivative of outside times derivative of inside

Inverse Function

d/dx[f⁻¹(x)] = 1/f'(f⁻¹(x))

One over derivative of original function

Logarithmic Diff.

d/dx[ln|f(x)|] = f'(x)/f(x)

Useful for products and quotients

Implicit Differentiation

d/dx[y] = dy/dx

Differentiate both sides, solve for dy/dx

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Trigonometric Derivatives

Sine

d/dx[sin(x)] = cos(x)

Derivative of sine is cosine

Cosine

d/dx[cos(x)] = -sin(x)

Derivative of cosine is negative sine

Tangent

d/dx[tan(x)] = sec²(x)

Derivative of tangent is secant squared

Secant

d/dx[sec(x)] = sec(x)tan(x)

Derivative of secant

Cosecant

d/dx[csc(x)] = -csc(x)cot(x)

Derivative of cosecant

Cotangent

d/dx[cot(x)] = -csc²(x)

Derivative of cotangent

Arcsine

d/dx[sin⁻¹(x)] = 1/√(1-x²)

Derivative of inverse sine

Arctangent

d/dx[tan⁻¹(x)] = 1/(1+x²)

Derivative of inverse tangent

∫

Basic Integrals

Constant Rule

∫c dx = cx + C

Integral of constant

Power Rule

∫xⁿ dx = xⁿ⁺¹/(n+1) + C

Add 1 to power, divide by new power

Exponential

∫eˣ dx = eˣ + C

Integral of exponential function

Natural Log

∫(1/x) dx = ln|x| + C

Integral gives natural logarithm

General Exponential

∫aˣ dx = aˣ/ln(a) + C

Exponential with arbitrary base

Sine

∫sin(x) dx = -cos(x) + C

Integral of sine

Cosine

∫cos(x) dx = sin(x) + C

Integral of cosine

Secant Squared

∫sec²(x) dx = tan(x) + C

Integral of secant squared

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Integration Techniques

Substitution

∫f(g(x))g'(x) dx = ∫f(u) du

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

Integration by Parts

∫u dv = uv - ∫v du

Useful for products of functions

Partial Fractions

∫[P(x)/Q(x)] dx

Decompose rational functions

Trigonometric Sub.

Use sin, cos, tan substitutions

For expressions with √(a²-x²), etc.

Fundamental Theorem

∫ₐᵇ f(x) dx = F(b) - F(a)

Connection between derivatives and integrals

Mean Value Theorem

∫ₐᵇ f(x) dx = f(c)(b-a)

For some c in [a,b]

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Limits

Basic Limit

lim[x→a] f(x) = L

Function approaches L as x approaches a

Limit Laws

lim[f(x) ± g(x)] = lim f(x) ± lim g(x)

Limits of sums and differences

Product Limit

lim[f(x)·g(x)] = lim f(x) · lim g(x)

Limit of product

Quotient Limit

lim[f(x)/g(x)] = lim f(x) / lim g(x)

Limit of quotient (if denominator ≠ 0)

L'Hôpital's Rule

lim[f(x)/g(x)] = lim[f'(x)/g'(x)]

For indeterminate forms 0/0 or ∞/∞

Squeeze Theorem

If g(x) ≤ f(x) ≤ h(x) and lim g(x) = lim h(x) = L, then lim f(x) = L

Bounded function theorem

Special Limit

lim[x→0] (sin x)/x = 1

Important trigonometric limit

Exponential Limit

lim[x→∞] (1 + 1/x)ˣ = e

Definition of e

∞

Series & Sequences

Geometric Series

Σarⁿ = a/(1-r) for |r| < 1

Sum of infinite geometric series

p-Series

Σ(1/nᵖ) converges if p > 1

Convergence test for p-series

Taylor Series

f(x) = Σ[f⁽ⁿ⁾(a)/n!](x-a)ⁿ

Function expansion around point a

Maclaurin Series

f(x) = Σ[f⁽ⁿ⁾(0)/n!]xⁿ

Taylor series centered at 0

Ratio Test

lim|aₙ₊₁/aₙ| = L

Converges if L < 1, diverges if L > 1

Root Test

lim|aₙ|^(1/n) = L

Converges if L < 1, diverges if L > 1

Integral Test

Σaₙ and ∫f(x)dx both converge or both diverge

For positive, decreasing sequences

Power Series

Σaₙ(x-c)ⁿ

Series in powers of (x-c)

∞Common Taylor/Maclaurin Series

eˣ

1 + x + x²/2! + x³/3! + ...

sin(x)

x - x³/3! + x⁵/5! - x⁷/7! + ...

cos(x)

1 - x²/2! + x⁴/4! - x⁶/6! + ...

ln(1+x)

x - x²/2 + x³/3 - x⁴/4 + ...

1/(1-x)

1 + x + x² + x³ + ...

(1+x)ⁿ

1 + nx + n(n-1)x²/2! + ...

Integration Strategy Guide

When to Use Each Technique

Substitution

When you see f(g(x)) · g'(x)

Integration by Parts

For products: polynomials × (trig, exp, ln)

Partial Fractions

For rational functions (polynomial/polynomial)

Trigonometric Substitution

For √(a²-x²), √(a²+x²), √(x²-a²)

Common Substitutions

√(a²-x²)x = a sin θ

√(a²+x²)x = a tan θ

√(x²-a²)x = a sec θ

∫eᵃˣ dxeᵃˣ/a + C

∫sin(ax) dx-cos(ax)/a + C

∫1/(x²+a²) dx(1/a)tan⁻¹(x/a) + C

Series Convergence Tests

Test	Condition	Conclusion	Best For
Divergence Test	lim aₙ ≠ 0	Diverges	First test to try
Ratio Test	lim \|aₙ₊₁/aₙ\| = L	L < 1: converges, L > 1: diverges	Factorials, exponentials
Root Test	lim \|aₙ\|^(1/n) = L	L < 1: converges, L > 1: diverges	nth powers
Comparison	0 ≤ aₙ ≤ bₙ	If Σbₙ converges, then Σaₙ converges	When you know similar series
Integral Test	f positive, decreasing	Σaₙ and ∫f(x)dx same behavior	p-series, continuous functions

About Calculus

Calculus is the mathematical study of continuous change, developed independently by Newton and Leibniz in the 17th century. It consists of two main branches: differential calculus (derivatives) which deals with rates of change and slopes, and integral calculus which deals with accumulation of quantities and areas under curves.

This comprehensive calculus reference covers the fundamental formulas and techniques from single-variable calculus through advanced topics like series and convergence tests. From basic derivative rules to complex integration techniques, these tools are essential for understanding and applying calculus in science, engineering, economics, and beyond.

Whether you're a student learning calculus for the first time, an engineer applying calculus to real-world problems, or a researcher using advanced mathematical techniques, this reference provides quick access to the most important formulas and relationships you'll encounter in calculus.

How to Apply Calculus Formulas

Identify the Problem Type

Determine if you need to find a derivative, integral, limit, or analyze a series.

Choose the Right Technique

Select the appropriate rule or method based on the function's structure and complexity.

Apply Systematically

Work step-by-step, showing all intermediate steps and checking your work along the way.

Verify and Interpret

Check your answer by differentiation/integration and interpret the result in context.

Calculus Study Tips

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Master the Fundamentals

Ensure you understand limits before derivatives, and derivatives before integrals.

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Practice Recognition

Learn to quickly identify which technique to use for each type of problem.

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Check by Differentiation

Verify integrals by taking the derivative of your answer.

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Visualize Functions

Graph functions to understand behavior and verify analytical results.

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Learn Common Patterns

Memorize derivatives and integrals of common functions.

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Practice Applications

Work on word problems to understand real-world applications of calculus.

Frequently Asked Questions

Get answers to common questions about calculus reference