Differential calculation

The foundations of differential calculus, the introduction of the notion of derivative, the operating rules for derivatives and the link between integration and derivation, conceived as operations that are inverses of each other, date back to the seventeenth century, mainly to Newton (1642-1727) and Leibniz (1647-1716). It was the latter mathematician who introduced the notation dy/dx defining the derivative of a function y.

Rolle's theorem (1652-1719) dates from 1691, and L'Hospital's rule from 1696. In 1715, Taylor (1685-1731) formulated the formula that bears his name. Taylor's formulas with Lagrange remainder and integral remainder were rigorously demonstrated by Lagrange (1736-1813).

Multivariate differential calculus appeared in the first half of the 18th century. In connection with physical problems (mechanics, hydrodynamics), the first partial differential equations appeared. In 1743, d'Alembert (1717-1783) studied the equation for the oscillations of a heavy chain. In 1746, he wrote the equation for vibrating strings (∂ ²y/ ∂t² = ∂ ²y/ ∂x ²), which he solved a few years later.

Laplace (1749-1827), following on from his work in astronomy, also took an interest in partial differential equations and attempted a first theory of second-order linear equations.

Throughout the 19th century, mathematicians helped to clarify differential calculus and give it its modern vigor, while the study of differential and partial differential equations remains as relevant as ever.