Mathematical Formulas

A collection of mathematical formulas organized by category.

Algebra

Basic algebraic formulas and identities

Quadratic Formula

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Difference of Squares

a2−b2=(a+b)(a−b)a^2 - b^2 = (a+b)(a-b)

Square of Binomial

(a+b)2=a2+2ab+b2(a+b)^2 = a^2 + 2ab + b^2

Cube of Binomial

(a+b)3=a3+3a2b+3ab2+b3(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3

Sum of Cubes

a3+b3=(a+b)(a2−ab+b2)a^3 + b^3 = (a+b)(a^2-ab+b^2)

Difference of Cubes

a3−b3=(a−b)(a2+ab+b2)a^3 - b^3 = (a-b)(a^2+ab+b^2)

Calculus

Differentiation, integration, and related concepts

Power Rule (Differentiation)

ddx[xn]=nxn−1\frac{d}{dx}[x^n] = nx^{n-1}

Product Rule

ddx[f(x)g(x)]=f′(x)g(x)+f(x)g′(x)\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)

Quotient Rule

ddx[f(x)g(x)]=f′(x)g(x)−f(x)g′(x)[g(x)]2\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}

Chain Rule

ddx[f(g(x))]=f′(g(x))g′(x)\frac{d}{dx}[f(g(x))] = f'(g(x))g'(x)

Power Rule (Integration)

∫xndx=xn+1n+1+C,n≠−1\int x^n dx = \frac{x^{n+1}}{n+1} + C, \quad n \neq -1

Exponential Integration

∫exdx=ex+C\int e^x dx = e^x + C

Logarithmic Integration

∫1xdx=ln⁡∣x∣+C\int \frac{1}{x} dx = \ln|x| + C

Geometry

Formulas for areas, volumes, and properties of shapes

Circle Area

A=πr2A = \pi r^2

Circle Circumference

C=2πrC = 2\pi r

Sphere Volume

V=43πr3V = \frac{4}{3}\pi r^3

Sphere Surface Area

A=4πr2A = 4\pi r^2

Cylinder Volume

V=πr2hV = \pi r^2 h

Cone Volume

V=13πr2hV = \frac{1}{3}\pi r^2 h

Pythagorean Theorem

a2+b2=c2a^2 + b^2 = c^2

Trigonometry

Trigonometric identities and formulas

Sine and Cosine Relation

sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1

Tangent Definition

tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}

Sine Addition

sin⁡(α+β)=sin⁡αcos⁡β+cos⁡αsin⁡β\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta

Cosine Addition

cos⁡(α+β)=cos⁡αcos⁡β−sin⁡αsin⁡β\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta

Double Angle (Sine)

sin⁡2θ=2sin⁡θcos⁡θ\sin 2\theta = 2\sin \theta \cos \theta

Double Angle (Cosine)

cos⁡2θ=cos⁡2θ−sin⁡2θ=2cos⁡2θ−1=1−2sin⁡2θ\cos 2\theta = \cos^2 \theta - \sin^2 \theta = 2\cos^2 \theta - 1 = 1 - 2\sin^2 \theta

Law of Sines

asin⁡A=bsin⁡B=csin⁡C\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}

Law of Cosines

c2=a2+b2−2abcos⁡Cc^2 = a^2 + b^2 - 2ab\cos C

Statistics & Probability

Statistical formulas and probability concepts

Arithmetic Mean

xˉ=1n∑i=1nxi\bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i

Variance

σ2=1n∑i=1n(xi−xˉ)2\sigma^2 = \frac{1}{n}\sum_{i=1}^{n} (x_i - \bar{x})^2

Standard Deviation

σ=1n∑i=1n(xi−xˉ)2\sigma = \sqrt{\frac{1}{n}\sum_{i=1}^{n} (x_i - \bar{x})^2}

Normal Distribution

f(x)=1σ2πe−12(x−μσ)2f(x) = \frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{1}{2}(\frac{x-\mu}{\sigma})^2}

Binomial Coefficient

(nk)=n!k!(n−k)!\binom{n}{k} = \frac{n!}{k!(n-k)!}

Bayes' Theorem

P(A∣B)=P(B∣A)P(A)P(B)P(A|B) = \frac{P(B|A)P(A)}{P(B)}

Correlation Coefficient

r=∑i=1n(xi−xˉ)(yi−yˉ)∑i=1n(xi−xˉ)2∑i=1n(yi−yˉ)2r = \frac{\sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum_{i=1}^{n} (x_i - \bar{x})^2 \sum_{i=1}^{n} (y_i - \bar{y})^2}}

Linear Algebra

Matrices, determinants, and vector operations

Matrix Multiplication

Cij=∑k=1nAikBkjC_{ij} = \sum_{k=1}^{n} A_{ik}B_{kj}

2×2 Determinant

det⁡(abcd)=ad−bc\det\begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc

Dot Product

a⃗⋅b⃗=∑i=1naibi=∣a⃗∣∣b⃗∣cos⁡θ\vec{a} \cdot \vec{b} = \sum_{i=1}^{n} a_i b_i = |\vec{a}||\vec{b}|\cos\theta

Cross Product (3D)

a⃗×b⃗=(a2b3−a3b2,a3b1−a1b3,a1b2−a2b1)\vec{a} \times \vec{b} = (a_2b_3 - a_3b_2, a_3b_1 - a_1b_3, a_1b_2 - a_2b_1)

Eigenvalue Equation

Av⃗=λv⃗A\vec{v} = \lambda\vec{v}

Matrix Trace

tr(A)=∑i=1nAii\text{tr}(A) = \sum_{i=1}^{n} A_{ii}

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