Math explainers · 9 pages
Advanced math explainers
Clear, citable explainers for the core concepts of advanced mathematics and calculus — derivative rules, product/quotient/chain rules, taylor series, and asymptotes.
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All 9 advanced math explainers pages
Quotient rule
The quotient rule is a formula for finding the derivative of a quotient of two differentiable functions. For a function f(x) = u(x) / v(x), the derivative is f'(x) = (u'(x)v(x) - u(x)v'(x)) / (v(x))², provided v(x) is not zero. In word terms, it is the derivative of the top times the bottom, minus the top times the derivative of the bottom, all divided by the bottom squared.
DefinitionChain rule
The chain rule is a formula for differentiating composite functions, where one function is nested inside another. If y = f(g(x)), then its derivative with respect to x is dy/dx = f'(g(x)) · g'(x). In Leibniz notation, if y = f(u) and u = g(x), then dy/dx = (dy/du) · (du/dx), indicating that the rates of change of the nested functions multiply.
DefinitionProduct rule
The product rule is a formula for finding the derivative of a product of two differentiable functions. For a function f(x) = u(x)v(x), the derivative is f'(x) = u'(x)v(x) + u(x)v'(x). In simple terms, it is the derivative of the first function times the second, plus the first function times the derivative of the second.
DefinitionTaylor series
A Taylor series is an infinite polynomial approximation of a function near a specific point. For a function f(x) that is infinitely differentiable at a point a, the Taylor series is defined as the sum from n = 0 to infinity of (f⁽ⁿ⁾(a) / n!) · (x - a)ⁿ. When the series is centred at zero (a = 0), it is specifically called a Maclaurin series.
DefinitionAsymptotes
An asymptote is a straight line that a curve approaches arbitrarily closely as it moves towards infinity. Asymptotes are classified into three types: vertical, where the function values grow without bound; horizontal, which describes the long-term behaviour as x approaches infinity; and oblique (slant), where the curve approaches a diagonal line y = mx + c.
DefinitionDerivative rules
Derivative rules are fundamental calculus shortcut formulas that simplify the process of differentiation. Instead of using the formal limit definition of a derivative, these rules — including the power rule, product rule, quotient rule, and chain rule — provide algebraic shortcuts for finding the rates of change of polynomial, rational, trigonometric, and exponential functions.
DefinitionCalculus
Calculus is the mathematical study of continuous change. It is divided into two main branches: differential calculus, which studies rates of change and the slopes of curves (derivatives), and integral calculus, which studies the accumulation of quantities and the areas under curves (integrals). These branches are linked by the Fundamental Theorem of Calculus.
DefinitionFibonacci sequence
The Fibonacci sequence is an integer sequence defined by the recurrence relation Fₙ = Fₙ₋₁ + Fₙ₋₂. Starting with 0 and 1, the sequence begins: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, and so on. The ratio between consecutive Fibonacci numbers approaches the golden ratio as the sequence goes to infinity.
DefinitionGolden ratio
The Golden Ratio exists when two quantities have a ratio equal to the ratio of their sum to the larger of the two quantities. Mathematically, for quantities a and b (where a > b), (a + b) / a = a / b = φ. The exact value of φ is (1 + √5) / 2, which is an irrational number approximately equal to 1.618033.
