Partial derivative

Mathematical statement

The x-partial derivative of a bivariate function f at (x,y) fixes y and is defined by the difference-quotient limit as nonzero h tends to 0; the y-partial is analogous. This slice differentiates quadratic polynomials symbolically and exactly, rather than replacing the limit with a single finite difference.

Representative formula (exact TeX)

D_i(f)

Assumptions

• Coefficients have fixed order [1,x,y,x²,x*y,y²]; exercises use integer coefficients and integer points.
• Polynomial slices are globally defined, but the existence of partials is not used to infer continuity or total differentiability of general functions.

Symbol scope

• D_i — coordinate partial derivative
• f — bivariate real-valued function

Intuition, definition, and misconceptions

The x-partial fixes y; the y-partial fixes x.

The coordinate gradient here is the ordered collection of coordinate partials.

A finite difference is an approximation or a sampled quotient, not automatically the partial derivative.

Do not vary both coordinates at once or use one nonzero-h quotient to claim the existence of a partial derivative.

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

• Multivariable real-valued function — Partials depend on declared coordinates and domains.
• Real derivatives at a point and finite difference quotients — Fixing the other coordinates leaves a one-variable derivative.
• Average rate of change between distinct inputs — A partial expresses instantaneous change in one coordinate direction.

Verification boundary

Existence of partials does not imply general continuity, total differentiability or interchangeability of mixed partials.

not_formalized

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

Next obligation

Add gradients, total derivatives and regularity conditions for higher partials.

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.