Maths Methods

Unit 3 — Calculus, probability, functions reference and solver

Differentiation

Power Rule

d/dx[xⁿ] = nxⁿ⁻¹

Chain Rule

d/dx[f(g(x))] = f'(g(x))·g'(x)

Product Rule

d/dx[f·g] = f'g + fg'

Quotient Rule

d/dx[f/g] = (f'g - fg')/g²

Exponential

d/dx[eˣ] = eˣ

Natural Log

d/dx[ln x] = 1/x

Sine

d/dx[sin x] = cos x

Cosine

d/dx[cos x] = -sin x

Integration

Power Rule

∫xⁿ dx = xⁿ⁺¹/(n+1) + C, n ≠ -1

Exponential

∫eˣ dx = eˣ + C

Reciprocal

∫(1/x) dx = ln|x| + C

Sine

∫sin x dx = -cos x + C

Cosine

∫cos x dx = sin x + C

Definite Integral

∫ₐᵇ f(x) dx = F(b) - F(a)

Probability & Statistics

Pr(A ∪ B)

Pr(A) + Pr(B) - Pr(A ∩ B)

Conditional

Pr(A|B) = Pr(A ∩ B) / Pr(B)

Normal Distribution

X ~ N(μ, σ²)

Standard Normal

Z = (X - μ) / σ

Sample Proportion

p̂ = x/n

95% CI

p̂ ± 1.96√(p̂(1-p̂)/n)

Functions & Relations

Quadratic Formula

x = (-b ± √(b²-4ac)) / 2a

Discriminant

Δ = b² - 4ac

Inverse Function

f⁻¹(f(x)) = x

Logarithm Laws

log(ab) = log a + log b, log(a/b) = log a - log b

Change of Base

logₐ b = log b / log a