Leibnizregelen

Leibnizregelen is a Dutch term used to refer to several rules and principles attributed to the 17th‑century mathematician and philosopher Gottfried Wilhelm Leibniz. In Dutch mathematical and logical literature, the phrase is used to describe rules that bear Leibniz’s name and played a role in the development of calculus and logical reasoning.

In calculus, the term commonly covers the Leibniz rule for differentiating products, known as the productregel. It states that if u and v are differentiable functions of x, then d/dx [u(x)v(x)] = u′(x)v(x) + u(x)v′(x). This rule is a fundamental tool in differentiation and reflects Leibniz’s early use of differential notation.

Another key example is the Leibniz rule for differentiation under the integral sign. If f(x,t) is suitably well behaved and a(x) and b(x) are differentiable, then d/dx ∫_{a(x)}^{b(x)} f(x,t) dt = f(x,b(x)) b′(x) − f(x,a(x)) a′(x) + ∫_{a(x)}^{b(x)} ∂f/∂x (x,t) dt. This rule explains how differentiation and integration interact when limits depend on the differentiation variable.

In logic, Leibniz’s law or principle of the identity of indiscernibles is sometimes grouped under the broader heading of Leibnizregelen. It asserts that if x = y, then any property P holds for x if and only if it holds for y, governing the substitutivity of identicals in reasoning.

Usage of the term is mainly historical or pedagogical in Dutch and emphasizes Leibniz’s influence on multiple areas of mathematics and philosophy rather than a single theorem.