§ 01What Is Calculus?
Calculus is the mathematics of change. It's what you reach for once ordinary algebra can't quite describe how something is varying, accumulating, or interacting anymore.
Nearly every idea in the subject grows out of one question: what happens to a ratio or a sum as some quantity gets pushed closer and closer to a limiting value? Differentiation answers that for instantaneous rates of change: the slope of a curve at a single point, the velocity of a particle at a single instant. Integration answers it for accumulation instead, giving you the area under a curve or the total distance covered from a changing velocity. Differential equations then put both together to model systems that evolve over time.
This hub page is the map. Each section below links to a full lesson with explanations, worked examples, and an interactive quiz, so you can move through the subject one topic at a time.
§ 02The Learning Path
Calculus builds in a fairly strict order. This is the sequence most courses (and this site) follow.
§ 03The Five Pillars
Every topic below is a full hub in its own right, with lessons, worked examples, and quizzes.
Limits
What it means for a function to approach a value, one-sided limits, limits at infinity, indeterminate forms, continuity, the epsilon-delta definition, and L'Hôpital's rule.
Differentiation
Derivatives from first principles through the product, quotient, and chain rules, implicit and parametric differentiation, higher-order derivatives, and derivatives of trig, log, and exponential functions.
Applications of Differentiation
Putting derivatives to work: curve sketching, maxima and minima, optimisation, related rates, linearisation, Newton's method, the Mean Value Theorem, and Rolle's Theorem.
Integration
Riemann sums and the Fundamental Theorem of Calculus, substitution, integration by parts, partial fractions, trig integrals and substitution, improper integrals, arc length, and volumes of revolution.
Differential Equations
Ordinary differential equations from first-order linear and separable equations through second-order ODEs, undetermined coefficients, variation of parameters, and Laplace transforms.
§ 04Why Calculus Matters
Calculus isn't just an academic hurdle, it's the language used to describe almost every system that changes over time or space.
Nearly every application of calculus is a variation on one of these two questions, or both at once.
In physics, derivatives give velocity and acceleration from position, and integrals recover position and distance from velocity. In engineering, differential equations model everything from circuits to structural vibration. In economics, marginal cost and marginal revenue are derivatives, and consumer surplus is an integral. In biology, population models are differential equations; in statistics, probability densities are integrated to get probabilities. Wherever something changes continuously, calculus is the tool that makes it precise.
§ 05Beyond Core Calculus
Once the five pillars are solid, these related areas extend the same ideas further.
Series & Sequences
Convergence tests, power series, Taylor and Maclaurin series.
Multivariable Calculus
Partial derivatives, multiple integrals, and vector calculus.
Linear Algebra
Vectors, matrices, eigenvalues, and linear transformations.
Probability
Distributions built directly on integration and series.
Statistics
Inference methods that rely on calculus under the hood.