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  "body": "\n## Summary\n\nA derivation is a linear map on an algebra that satisfies the product rule $D(ab) = D(a) b + a D(b)$, see the [[definition|R1F0HCCE6F]].\nIt is the algebraic form of the [[derivative|0V1EJCT8G6]], and it makes sense without limits.\nDerivations are basic in [[differential algebra|GA5PTQGC6G]], algebraic geometry and Lie theory.\nThe derivations of a Lie algebra obey the same rule for the Lie bracket.\nThe commutator of two derivations is again a derivation, so the derivations of an algebra form a Lie algebra.\n\n## Explanations\n\n### L1 human\n\nThe derivative of a product obeys the product rule: $(fg)' = f'g + fg'$.\nA derivation is any map that adds up linearly and obeys this rule.\nFor polynomials, $d/dX$ is a derivation: it sends $X^3$ to $3X^2$ without using limits.\nAnother example on polynomials in $X$ and $Y$ is $X \\, \\partial/\\partial Y$, which sends $Y^2$ to $2XY$.\n",
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