# Partial derivative

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

A partial derivative fixes every other coordinate and takes the one-variable derivative along one coordinate slice as h tends to zero.

mko-partial-derivative · definition · v0.10.0



## 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

[Partial derivative: deriving the partial-derivative formulas of a quadratic from the difference-quotient limit, and computing a gradient](https://ocme.evemisslab.com/en/derivations/derivation-mko-partial-derivative-01/)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](https://ocme.evemisslab.com/en/mko/mko-multivariable-real-function/) — Partials depend on declared coordinates and domains.
- [Real derivatives at a point and finite difference quotients](https://ocme.evemisslab.com/en/mko/mko-derivative/) — Fixing the other coordinates leaves a one-variable derivative.
- [Average rate of change between distinct inputs](https://ocme.evemisslab.com/en/mko/mko-rate-of-change/) — 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.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/partial-derivative.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-partial-derivative/index.md)