Higher-order derivative

Mathematical statement

The order n belongs to N_0 and every intermediate function must be differentiable again on the declared domain. Existence of the n-th derivative does not imply existence of the next. Exact polynomial families use falling-factorial coefficients and give zero derivatives above the polynomial degree. Finite-order or even all finite-order differentiability alone does not imply analyticity or Taylor-series convergence.

Representative formula (exact TeX)

D_n(f)

Assumptions

• Positive families use low-to-high integer polynomial coefficient arrays.
• The order and evaluation point are bounded safe integers.

Symbol scope

• D_n — n-th derivative operator
• f — function with recursively existing derivatives

Intuition, definition, and misconceptions

Order zero returns the original function.

Each derivative step requires the preceding function to be differentiable.

Polynomial derivatives above their degree are zero.

Do not infer the next derivative from the preceding one, or analyticity from existence of all finite derivatives.

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 — Begin with the first derivative.
• Differentiating polynomial sums, products, and compositions — Differentiate polynomial terms.
• Mathematical induction — Derivative order is recursive.

Verification boundary

Polynomial certificates in this slice are not a general smoothness/analyticity theorem.

not_formalized

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

Next obligation

Add Taylor polynomials.

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.