Real derivatives at a point and finite difference quotients

Mathematical statement

For f:R→R, its derivative at x is lim_(h→0,h≠0)(f(x+h)−f(x))/h when this finite two-sided limit exists. This pulse treats exact affine and degree-at-most-two polynomial derivatives and the corner of |t−c| at t=c, where distinct one-sided derivatives give no derivative.

Representative formula (exact TeX)

D(f,x)

Assumptions

• The full real domain, center value, and both neighborhoods are permitted; a finite step h must be nonzero.
• For f(t)=a2*t²+a1*t+a0, a2=0 is allowed: the model is degree at most two, not necessarily a genuine quadratic with nonzero leading coefficient.
• Its derivative is 2a2*x+a1; its finite quotient is that value plus a2*h. For |t−c|, the derivative is one above c and minus one below c; at c it does not exist, rather than being zero.

Symbol scope

• D — real derivative at a specified point
• f — specified total real function
• x — real derivative center

Intuition, definition, and misconceptions

The center value f(x) is defined, but h=0 does not permit dividing the difference by zero. Simplify for h≠0, then examine whether its limit exists.

In the degree-at-most-two model, finite quotient and derivative may differ by a2*h. Special equalities do not turn every finite quotient into a derivative. Constant and affine boundary cases are valid.

At the absolute-value center, positive and negative steps give quotients one and minus one, so there is no two-sided derivative. A formal library may totalize deriv with a default value at nondifferentiable points; that value does not establish derivative existence.

Common errors include naming a finite quotient a derivative, using a zero step, or treating a totalized API default as a derivative at a corner.

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

• Function limits on punctured real domains — Use the two-sided limit of nonzero-step quotients.
• Average rate of change between distinct inputs — A finite quotient is not an instantaneous derivative.
• Univariate polynomials over the rationals — Use the degree-at-most-two coefficient model.
• Absolute value — Retain the two affine sides of the corner.

Verification boundary

Declared algebraic models and corner decisions can be checked exactly; no new general differentiation or formal derivative Evidence is attached.

not_formalized

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

Next obligation

Bind HasDerivAt and nonzero-step slope limits, preserving existence versus totalized deriv.

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.