An algebraic derivation of an algebra over a field is a linear map satisfying the Leibniz rule for its multiplication: . This definition makes sense even when the multiplication is not associative. A derivation of an algebra uses an associative product; a derivation of a Lie algebra uses the Lie bracket as its product.
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In differential algebra, a derivation is a mathematical operator that satisfies certain linearity and product rule properties, similar to the way that derivatives function in calculus. More formally, a derivation on a differential ring (or differential algebra) is a mapping that associates to each element of the ring another element of the same ring, reflecting the idea of differentiation.