Alphabeta Math
PropositionStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 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.

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

Statement

For a commutative ring RR, polynomials f,g∈R[x]f,g\in R[x], and c∈Rc\in R:

  1. (f+g)′=f′+g′(f+g)'=f'+g' and (cf)′=cf′(cf)'=cf';
  2. (xn)′=nxn−1(x^n)'=n x^{n-1} for every positive nn, while every constant has derivative 00;
  3. (fg)′=f′g+fg′(fg)'=f'g+fg';
  4. if f′f' is nonzero, then deg⁡f′≤deg⁡f−1\deg f'\le\deg f-1.

Facts & Assumptions

Given: A commutative ring RR and polynomials f=∑iaixif=\sum_i a_i x^i and g=∑jbjxjg=\sum_j b_jx^j.

[L1]

The coefficient of xnx^n in f′f' is (n+1)an+1(n+1)a_{n+1} (The formal derivative of a polynomial).

[L2]

Finite sums may be split, reindexed, and summed in either order over finite products (Finite commutative-monoid sums are invariant under bijective reindexing, split over disjoint unions, and satisfy the finite Fubini rule).

Proof

technique · direct
1.1

Comparing the coefficient at each index in [L1] proves additivity and scalar linearity; applying [L1] to a monomial gives the power rule, including derivative 00 for constants.

givenL1algebra
1.2

The coefficient of xnx^n in (fg)′(fg)' is (n+1)∑i+j=n+1aibj=∑r+s=n((r+1)ar+1bs+ar(s+1)bs+1)(n+1)\sum_{i+j=n+1}a_i b_j=\sum_{r+s=n}((r+1)a_{r+1}b_s+a_r(s+1)b_{s+1}), where [L2] reindexes the two finite sums; these are exactly the coefficients of f′g+fg′f'g+fg'.

givenL1L2algebra
2.1

If ff has degree dd and f′≠0f'\ne0, then [L1] makes every coefficient of f′f' above index d−1d-1 zero, so deg⁡f′≤d−1\deg f'\le d-1; steps 1.1 and 1.2 establish all remaining claims.

step 1.1step 1.2L1∎

Depends on

Used by

Cited to discharge well-definedness by The formal derivative of a polynomial.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 37 results over 11 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