Differentiating polynomial sums, products, and compositions

Mathematical statement

For real functions differentiable at the relevant points, (f+g)′=f′+g′, (cf)′=cf′, and (fg)′=f′g+fg′. If g is differentiable at x and f at g(x), then (f∘g)′(x)=f′(g(x))g′(x). Here f,g are full-real-domain polynomials represented by ascending coefficients starting with the constant term.

Representative formula (exact TeX)

D(f,x)

Assumptions

• Input lists have one through four safe-integer coefficients, hence degree at most three. Normalize trailing zeros; an empty list is invalid.
• Input polynomials are differentiable everywhere on R. In a composition, evaluate the outer derivative at g(x), not x; these instances do not establish differentiability of arbitrary functions.
• Acceptance expands the sum, scalar multiple, product, or composition and differentiates it; the producer separately uses the rules. Resulting degrees may increase, so do not truncate new terms.

Symbol scope

• D — point value of a specified polynomial derivative
• f — combined or specified real polynomial function
• x — real evaluation point with integer exercise instances

Intuition, definition, and misconceptions

The product rule is not f′g′. The chain rule evaluates the outer derivative at the inner value and multiplies by the inner derivative. Remove constant terms only when truly irrelevant to the specified operation.

Ascending [a0,a1,a2] means a0+a1*x+a2*x², not descending powers. Appending trailing zeros represents the same polynomial, not a new question.

Retain zero scalars, zero polynomials, and constant outer or inner functions. A constant composite has derivative zero. Deduplication must not discard substantive scales or evaluation points.

Common errors include multiplying derivatives for a product, omitting the inner derivative or evaluating at the wrong point in a composition, or reversing coefficient order.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Real derivatives at a point and finite difference quotients — Retain derivative-existence semantics.
• Univariate polynomials over the rationals — Use ascending coefficients and finite operations.
• Function composition — The outer function receives g(x).

Verification boundary

Finite polynomial identities and values can be recomputed exactly; general differentiation-rule formal Evidence remains unestablished.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Bridge pinned polynomial derivative/eval and HasDerivAt.add/mul/comp, without replacing general differentiability premises by finite values.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.