Calculus is the mathematical study of continuous change, comprising differential and integral calculus.
Calculus is divided into two main branches. Differential calculus studies rates of change and slopes of curves — the derivative f'(x) gives the instantaneous rate of change of f at x. Integral calculus studies accumulation of quantities and areas under curves — the definite integral ∫ₐᵇ f(x) dx gives the net area between f(x) and the x-axis. The Fundamental Theorem of Calculus (FTC) connects these branches: Part 1 states that ddx ∫ₐˣ f(t) dt = f(x), and Part 2 states that ∫ₐᵇ f(x) dx = F(b) - F(a) where F' = f. Calculus extends to multivariable functions through partial derivatives and multiple integrals. Key applications: physics (motion, electromagnetism, thermodynamics), engineering (optimization, fluid dynamics), economics (marginal analysis, elasticity), biology (population dynamics, epidemiology), and machine learning (gradient descent, backpropagation). Techniques include substitution, integration by parts, L'Hôpital's rule, and Taylor series expansions.
Calculus was independently developed by Isaac Newton and Gottfried Wilhelm Leibniz in the 17th century. Newton developed his "method of fluxions" (1665-1666) to describe planetary motion and universal gravitation, but delayed publication. Leibniz (1684) published first and developed the superior notation (dy/dx for derivatives, ∫ for integrals) still used today, sparking a bitter priority dispute. Earlier precursors include Archimedes' method of exhaustion (250 BCE), Pierre de Fermat's work on tangents and maxima (1630s), and Bonaventura Cavalieri's method of indivisibles (1635). The rigorous foundations were established by Augustin-Louis Cauchy (1820s) using limits, and later by Karl Weierstrass with the ε-δ definition. In the 20th century, Abraham Robinson developed non-standard calculus using infinitesimals, and Henri Lebesgue created measure theory, generalizing integration.