DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 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 $f=\sum_{i\ge0}a_i x^i\in R[x]$ . Its formal derivative is

$f':=\sum_{i\ge1} i a_i x^{i-1}.$

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

Used by

Linearity, power rule, Leibniz rule and the degree bound for the formal derivative Proposition

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 11 results over 6 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Brian Conrad, Differential Criterion and Primitivity, Section 1 (standard reference, not scraped)