DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-08-11

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The formal derivative of a polynomial

Definition

Let $R$ be a commutative ring and let $f = \sum_{i \geq 0} a_{i} x^{i} \in R [x]$ . Its formal derivative is

$f^{'} : = \sum_{i \geq 1} i a_{i} x^{i - 1} .$

Here $i a_{i}$ means the sum of $i$ copies of $a_{i}$ in the additive group of $R$ . Equivalently, the coefficient of $x^{n}$ in $f^{'}$ is $(n + 1) a_{n + 1}$ . The sequence defining $f^{'}$ has finite support because the coefficient sequence of $f$ does (The polynomial ring over a commutative ring as finitely supported coefficient sequences with convolution). This is an algebraic operation and does not use a limit.

Depends on

The polynomial ring over a commutative ring as finitely supported coefficient sequences with convolution

Dependency tree · 2 levels

Sources

Brian Conrad, Differential Criterion and Primitivity, Section 1