The Ultimate Mathematics Formula Directory

1. Algebra & Fundamentals

Roots and Quadratics

• Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
• Vertex Formula (Parabola): $x = -\frac{b}{2a}$, $y = f(-\frac{b}{2a})$
• Difference of Squares: $a^2 - b^2 = (a-b)(a+b)$
• Perfect Square Trinomial: $a^2 \pm 2ab + b^2 = (a \pm b)^2$
• Sum/Difference of Cubes: $a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)$

Exponent and Radical Laws

• $x^a \cdot x^b = x^{a+b}$
• $\frac{x^a}{x^b} = x^{a-b}$
• $(x^a)^b = x^{ab}$
• $x^{-a} = \frac{1}{x^a}$
• $x^{\frac{a}{b}} = \sqrt[b]{x^a}$
• $\sqrt{ab} = \sqrt{a}\sqrt{b}$

Logarithmic Properties

• Definition: $y = \log_b(x) \iff b^y = x$
• Product: $\log_b(xy) = \log_b(x) + \log_b(y)$
• Quotient: $\log_b(\frac{x}{y}) = \log_b(x) - \log_b(y)$
• Power: $\log_b(x^n) = n\log_b(x)$
• Change of Base: $\log_b(x) = \frac{\ln(x)}{\ln(b)}$
• Natural Log Inverse: $\ln(e^x) = x$ and $e^{\ln(x)} = x$

2. Sequences and Series

• Arithmetic Sequence (nth term): $a_n = a_1 + (n-1)d$
• Arithmetic Series (Sum): $S_n = \frac{n}{2}(a_1 + a_n)$
• Geometric Sequence (nth term): $a_n = a_1r^{n-1}$
• Geometric Series (Finite Sum): $S_n = \frac{a_1(1-r^n)}{1-r}$
• Geometric Series (Infinite Sum): $S_\infty = \frac{a_1}{1-r} \quad (\text{if } |r| < 1)$
• Taylor Series Expansion: $f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n$

3. Trigonometry & Geometry

Core Identities

• Pythagorean 1: $\sin^2(x) + \cos^2(x) = 1$
• Pythagorean 2: $1 + \tan^2(x) = \sec^2(x)$
• Pythagorean 3: $1 + \cot^2(x) = \csc^2(x)$
• Reciprocals: $\csc(x) = \frac{1}{\sin(x)}$, $\sec(x) = \frac{1}{\cos(x)}$, $\cot(x) = \frac{1}{\tan(x)}$

Advanced Angle Formulas

• Double Angle (Sine): $\sin(2x) = 2\sin(x)\cos(x)$
• Double Angle (Cosine): $\cos(2x) = \cos^2(x) - \sin^2(x) = 2\cos^2(x) - 1 = 1 - 2\sin^2(x)$
• Half Angle (Sine): $\sin^2(x) = \frac{1 - \cos(2x)}{2}$
• Half Angle (Cosine): $\cos^2(x) = \frac{1 + \cos(2x)}{2}$
• Sum/Difference: $\sin(x \pm y) = \sin(x)\cos(y) \pm \cos(x)\sin(y)$

Geometry

• Circle: Area $= \pi r^2$, Circumference $= 2\pi r$
• Sphere: Volume $= \frac{4}{3}\pi r^3$, Surface Area $= 4\pi r^2$
• Cylinder: Volume $= \pi r^2 h$, Surface Area $= 2\pi rh + 2\pi r^2$
• Cone: Volume $= \frac{1}{3}\pi r^2 h$

4. Differential Calculus (Derivatives)

The Fundamental Rules

• Limit Definition: $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$
• Power Rule: $\frac{d}{dx}[x^n] = nx^{n-1}$
• Product Rule: $\frac{d}{dx}[uv] = u'v + uv'$
• Quotient Rule: $\frac{d}{dx}[\frac{u}{v}] = \frac{u'v - uv'}{v^2}$
• Chain Rule: $\frac{d}{dx}[f(g(x))] = f'(g(x))g'(x)$

Transcendental Derivatives

• Exponentials: $\frac{d}{dx}[e^x] = e^x$, $\frac{d}{dx}[a^x] = a^x \ln(a)$
• Logarithms: $\frac{d}{dx}[\ln(x)] = \frac{1}{x}$, $\frac{d}{dx}[\log_a(x)] = \frac{1}{x \ln(a)}$
• Sine/Cosine: $\frac{d}{dx}[\sin(x)] = \cos(x)$, $\frac{d}{dx}[\cos(x)] = -\sin(x)$
• Tangent/Cotangent: $\frac{d}{dx}[\tan(x)] = \sec^2(x)$, $\frac{d}{dx}[\cot(x)] = -\csc^2(x)$
• Secant/Cosecant: $\frac{d}{dx}[\sec(x)] = \sec(x)\tan(x)$, $\frac{d}{dx}[\csc(x)] = -\csc(x)\cot(x)$

Inverse Trigonometric Derivatives

• $\frac{d}{dx}[\arcsin(x)] = \frac{1}{\sqrt{1 - x^2}}$
• $\frac{d}{dx}[\arccos(x)] = -\frac{1}{\sqrt{1 - x^2}}$
• $\frac{d}{dx}[\arctan(x)] = \frac{1}{1 + x^2}$

5. Integral Calculus (Antiderivatives)

Fundamental Rules & Theorems

• Power Rule: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)$ Solve Power Rule Integral
• The 1/x Exception: $\int \frac{1}{x} \, dx = \ln|x| + C$ Solve 1/x Integral
• Integration by Parts: $\int u \, dv = uv - \int v \, du$
• Fundamental Theorem of Calculus: $\int_a^b f(x) \, dx = F(b) - F(a)$

Trigonometric Integrals

• $\int \sin(x) \, dx = -\cos(x) + C$ Solve Sine Integral
• $\int \cos(x) \, dx = \sin(x) + C$ Solve Cosine Integral
• $\int \sec^2(x) \, dx = \tan(x) + C$ Solve Secant Squared Integral
• $\int \sec(x)\tan(x) \, dx = \sec(x) + C$
• $\int \tan(x) \, dx = -\ln|\cos(x)| + C$ or $\ln|\sec(x)| + C$
• $\int \sec(x) \, dx = \ln|\sec(x) + \tan(x)| + C$

Advanced Trigonometric Substitution Forms

When confronting radicals, use these exact substitutions:

• For $\sqrt{a^2 - x^2}$: Substitute $x = a\sin(\theta)$, use $1 - \sin^2(\theta) = \cos^2(\theta)$.
• For $\sqrt{a^2 + x^2}$: Substitute $x = a\tan(\theta)$, use $1 + \tan^2(\theta) = \sec^2(\theta)$.
• For $\sqrt{x^2 - a^2}$: Substitute $x = a\sec(\theta)$, use $\sec^2(\theta) - 1 = \tan^2(\theta)$.

6. Limits & Continuity

Limit Properties

• Sum/Difference: $\lim_{x \to a} [f(x) \pm g(x)] = \lim_{x \to a} f(x) \pm \lim_{x \to a} g(x)$
• Product: $\lim_{x \to a} [f(x)g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)$
• Quotient: $\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}$ (if denominator $\neq 0$)

Important Limit Theorems

• Squeeze Theorem: If $f(x) \leq g(x) \leq h(x)$ for all $x$ near $a$, and $\lim_{x \to a} f(x) = \lim_{x \to a} h(x) = L$, then $\lim_{x \to a} g(x) = L$.
• L'Hôpital's Rule: If $\lim \frac{f(x)}{g(x)}$ is of the form $\frac{0}{0}$ or $\frac{\infty}{\infty}$, then $\lim \frac{f(x)}{g(x)} = \lim \frac{f'(x)}{g'(x)}$.
• Special Trig Limit 1: $\lim_{x \to 0} \frac{\sin(x)}{x} = 1$ Solve Sine Limit
• Special Trig Limit 2: $\lim_{x \to 0} \frac{1 - \cos(x)}{x} = 0$ Solve Cosine Limit

7. Applications of Derivatives

Theorems and Approximations

• Mean Value Theorem (MVT): If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, there exists a $c$ in $(a,b)$ such that $f'(c) = \frac{f(b)-f(a)}{b-a}$.
• Rolle's Theorem: If MVT conditions hold and $f(a) = f(b)$, then $f'(c) = 0$.
• Linear Approximation: $L(x) = f(a) + f'(a)(x-a)$
• Newton's Method: $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$

Curve Sketching Rules

• Critical Points: Where $f'(x) = 0$ or is undefined.
• Increasing/Decreasing: $f$ is increasing if $f'(x) > 0$, decreasing if $f'(x) < 0$.
• Inflection Points: Where $f''(x)$ changes sign.
• Concavity: Concave up if $f''(x) > 0$, concave down if $f''(x) < 0$.
• First Derivative Test: If $f'(x)$ changes from $+$ to $-$ at $c$, $f(c)$ is a local maximum.

8. Applications of Integrals

Area and Volume

• Area Between Curves: $A = \int_a^b [f(x) - g(x)] \, dx$ (where $f(x) \geq g(x)$)
• Volume (Disk Method): $V = \pi \int_a^b [R(x)]^2 \, dx$
• Volume (Washer Method): $V = \pi \int_a^b ([R_{outer}(x)]^2 - [r_{inner}(x)]^2) \, dx$
• Volume (Shell Method): $V = 2\pi \int_a^b x f(x) \, dx$

Advanced Integral Formulas

• Arc Length: $L = \int_a^b \sqrt{1 + [f'(x)]^2} \, dx$
• Surface Area of Revolution: $S = 2\pi \int_a^b f(x) \sqrt{1 + [f'(x)]^2} \, dx$
• Average Value of a Function: $f_{avg} = \frac{1}{b-a} \int_a^b f(x) \, dx$

9. Differential Equations

• Separable DEs: $\frac{dy}{dx} = g(x)h(y) \implies \int \frac{1}{h(y)} \, dy = \int g(x) \, dx$
• Newton's Law of Cooling: $\frac{dT}{dt} = k(T - T_s) \implies T(t) = T_s + (T_0 - T_s)e^{kt}$
• Logistic Growth Model: $\frac{dP}{dt} = kP(1 - \frac{P}{K})$
• Logistic Solution: $P(t) = \frac{K}{1 + Ae^{-kt}}$ where $A = \frac{K - P_0}{P_0}$

10. Parametric, Polar, and Vector Calculus

Parametric Equations

• First Derivative: $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$
• Second Derivative: $\frac{d^2y}{dx^2} = \frac{\frac{d}{dt}[\frac{dy}{dx}]}{dx/dt}$
• Parametric Arc Length: $L = \int_{t_1}^{t_2} \sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2} \, dt$

Polar Coordinates

• Conversion: $x = r\cos(\theta)$, $y = r\sin(\theta)$, $r^2 = x^2 + y^2$, $\tan(\theta) = \frac{y}{x}$
• Polar Derivative: $\frac{dy}{dx} = \frac{r\cos(\theta) + r'\sin(\theta)}{-r\sin(\theta) + r'\cos(\theta)}$
• Area of a Polar Region: $A = \frac{1}{2} \int_{\alpha}^{\beta} [r(\theta)]^2 \, d\theta$