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# derivation

[der-uh-vey-shuhn]
derivation, in grammar: see inflection.

In logic and mathematics, an argument that establishes a proposition's validity. Formally, it is a finite sequence of formulas generated according to accepted rules. Each formula either is an axiom or is derived from a previously established theorem, and the last formula is the statement that is to be proven. The essence of deductive reasoning (see deduction), this is the basis of Euclidean geometry and all scientific methods inspired by it. An alternative form of proof, called mathematical induction, applies to propositions defined through processes based on the counting numbers. If the proposition holds for math.n = 1 and can be shown to hold for math.n = math.k + 1 whenever math.n = math.k (a constant) is also true, then it holds for all values of math.n. An example is the assertion that the sum of the first math.n counting numbers is math.n(math.n + 1)/2.

Derivation may refer to:

• Derivation (abstract algebra), a function on an algebra which generalizes certain features of the derivative operator
• Derivation (linguistics)
• Derivation in differential algebra, a unary function satisfying the Leibniz product law
• Formal proof or derivation, a sequence of sentences each of which is an axiom or follows from the preceding sentences in the sequence by a rule of inference
• Parse tree or concrete syntax tree, a tree that represents the syntactic structure of a string according to some formal grammar
• The creation of a derived row, in the twelve-tone musical technique
• An after-the-fact justification for an action, in the work of sociologist Vilfredo Pareto