A Limit-Free Algebraic-Geometric Construction of Derivatives for Elementary Functions

Davit Kapanadze

arxiv: 2605.20228 · v1 · pith:7LX7KRA6new · submitted 2026-05-15 · 🧮 math.GM

A Limit-Free Algebraic-Geometric Construction of Derivatives for Elementary Functions

Davit Kapanadze This is my paper

Pith reviewed 2026-05-21 09:01 UTC · model grok-4.3

classification 🧮 math.GM

keywords derivative constructionlimit-free approachalgebraic-geometric methodstangent line interpretationlocal linear structureelementary functionsinverse symmetry

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The pith

Derivatives of elementary functions can be derived geometrically from the tangent line without starting from limits.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper seeks to establish that derivatives for rational powers, exponentials, logarithms, trigonometric functions, and inverse trigonometric functions can be constructed using the geometric meaning of the tangent line, inverse symmetry, and local linear structure. It defines the derivative from the start as the slope of the tangent at a point and shows how the usual rules follow from these interconnected structures. This would matter to a reader interested in foundations because it reverses the typical order by introducing tangent and linearity before formal limits, yet arrives at the same results as standard calculus.

Core claim

By treating the derivative as the slope coefficient of the tangent line at each point and employing local linear structure together with inverse symmetries, the differentiation rules for elementary functions emerge directly from algebraic and geometric relations, and these rules prove consistent with those obtained through the standard limit definition of the derivative.

What carries the argument

The tangent line interpreted geometrically as the carrier of the local slope, extended by local linear approximations and inverse function relations to generate explicit derivative formulas.

If this is right

The standard differentiation rules for rational powers, exponentials, logarithms, trigonometric, and inverse trigonometric functions follow from the geometric and algebraic structures described.
The derivative is introduced as a functional correspondence that assigns the slope of the tangent line to each point.
Classical formulas are shown to be consistent with limit-based analysis after the geometric construction.
The methodological sequence of tangent, local linear structure, and then limit formalization is proposed as a logical order.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This method could change how calculus is introduced by starting with geometric concepts before analytic ones.
The framework might be extended to show how other calculus properties, such as the chain rule, arise from the same local linear assumptions.
It suggests examining whether all elementary function properties can be derived in this order without presupposing limits.

Load-bearing premise

The assumption that the geometric interpretation of the tangent line and the local linear structure can be made rigorous enough to derive all the differentiation rules for elementary functions on their own, separate from limits.

What would settle it

Deriving the derivative of the exponential function solely through local linear structure and inverse symmetry, then verifying whether the result equals the function itself, would confirm or refute the construction if it fails to match the known value.

read the original abstract

This paper continues the author's previous work on a limit-free algebraic-geometric construction of the derivative in the class of polynomial functions and extends the proposed framework to elementary functions. Derivatives of rational power, exponential, logarithmic, trigonometric, and inverse trigonometric functions are constructed through the geometric interpretation of the tangent line, inverse symmetry, and local linear structure, without treating the limit as the initial defining mechanism. Within the proposed approach, the derivative is introduced from the outset as a functional correspondence assigning to each point the slope coefficient of the tangent line. The paper demonstrates that the classical differentiation formulas arise naturally from interconnected geometric and algebraic structures and are subsequently consistent with standard limit-based analysis. From a methodological perspective, the study proposes the logical sequence: Tangent, Local Linear Structure, Limit formalisation. Thus, the paper presents a conceptual bridge between geometric intuition, algebraic construction, and classical differential calculus.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Extends the author's polynomial work with geometric tangents for elementary functions but the independence from limits looks shaky for trig and exp.

read the letter

The main point is that this paper takes the author's earlier limit-free approach for polynomials and tries to carry it over to rational powers, exponentials, logs, trig, and inverse trig functions by starting from the tangent line and local linear structure instead of the difference quotient limit. It claims the usual derivative rules then drop out naturally from those geometric and algebraic relations, with the limit version coming later as a consistency check. The proposed sequence is Tangent first, then local linearity, then formal limits. For the polynomial part this kind of construction can be made to work cleanly with algebraic manipulations alone. The extension to the other elementary functions is the actual new piece here, and the paper at least sketches how inverse symmetry and local linearity might pin down the tangent slopes for each case. That is worth noting as a coherent attempt to reorder the logical dependencies in calculus. The soft spots are exactly where the stress-test note flags them. For sine or the exponential, any definition of the tangent that relies on addition formulas, angle identities, or series expansions will typically have used limit arguments or equivalent properties upstream, so the claimed priority of the geometric tangent over the limit becomes hard to maintain. The abstract gives no sample derivations or explicit checks against external benchmarks, which leaves the circularity question open. Without those steps it is difficult to see whether the constructions are independent or simply arranged to reproduce the known answers. The work is internally consistent as a re-description of standard results in geometric language, but it does not appear to produce new theorems or resolve open questions. Readers who care about alternative foundations or geometric-first teaching of calculus might find it useful to think through. A referee could usefully test whether the non-polynomial cases really avoid presupposing the derivatives they claim to derive. I would send it to review rather than desk-reject so the independence claim gets a proper check.

Referee Report

2 major / 2 minor

Summary. The paper extends prior limit-free work on polynomials to elementary functions by defining the derivative from the outset as the slope of the tangent line, constructed via geometric notions of inverse symmetry and local linear structure. It claims that the standard differentiation rules for rational powers, exponentials, logarithms, trigonometric functions, and inverse trigonometric functions arise naturally from these interconnected algebraic-geometric structures, after which consistency with the classical limit definition is shown. The methodological contribution is the proposed sequence Tangent → Local Linear Structure → Limit formalisation.

Significance. A rigorously non-circular geometric-algebraic foundation for the derivative of elementary functions would be a notable contribution to the foundations of calculus, offering an alternative entry point that prioritizes geometric primitives over analytic limits and potentially clarifying the conceptual order in which differentiation rules are introduced.

major comments (2)

The central claim requires an independent characterization of the tangent line and local linear structure for each elementary function (e.g., sin(x) and exp(x)) that does not encode the difference-quotient limit. The manuscript must supply explicit algebraic steps in the relevant derivation sections showing how the slope coefficient is obtained from inverse symmetry or local linearity alone, without presupposing the known derivative value or using angle-addition identities that themselves rest on limit arguments.
For the trigonometric and exponential cases, the construction must demonstrate that the resulting slope matches the classical formulas only after the geometric definition is fixed, rather than by matching to known results. Without such a demonstration (for instance, in the sections treating sin(x) or e^x), the logical sequence Tangent → Local Linear Structure → Limit risks circularity.

minor comments (2)

Clarify the precise definition of 'local linear structure' when first introduced, including any algebraic axioms or geometric postulates used to fix the tangent slope for non-polynomial functions.
Add explicit worked examples for at least one inverse trigonometric function to illustrate how the inverse-symmetry argument produces the derivative formula.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for these insightful comments on ensuring the non-circularity of our geometric-algebraic framework. We address each major point below and outline revisions to strengthen the explicit derivations while preserving the proposed logical sequence.

read point-by-point responses

Referee: The central claim requires an independent characterization of the tangent line and local linear structure for each elementary function (e.g., sin(x) and exp(x)) that does not encode the difference-quotient limit. The manuscript must supply explicit algebraic steps in the relevant derivation sections showing how the slope coefficient is obtained from inverse symmetry or local linearity alone, without presupposing the known derivative value or using angle-addition identities that themselves rest on limit arguments.

Authors: We agree that greater explicitness is needed to demonstrate independence from limit-based encodings. In the revised manuscript we will expand the relevant sections with step-by-step algebraic derivations: for sin(x) the tangent is characterized solely via inverse symmetry with respect to the point on the unit circle, yielding the slope coefficient directly from the resulting linear equation without angle-addition formulas; for exp(x) local linearity is fixed by the multiplicative functional equation translated into an algebraic condition on increments, again producing the slope without presupposing its classical value. These additions will make the extraction of the coefficient from geometric primitives fully transparent. revision: yes
Referee: For the trigonometric and exponential cases, the construction must demonstrate that the resulting slope matches the classical formulas only after the geometric definition is fixed, rather than by matching to known results. Without such a demonstration (for instance, in the sections treating sin(x) or e^x), the logical sequence Tangent → Local Linear Structure → Limit risks circularity.

Authors: The manuscript already derives the slope coefficients from the geometric definitions first and only afterward verifies consistency with the classical limit expressions. To address the concern directly we will insert a new subsection in each case (sin(x) and exp(x)) that explicitly records the geometric slope obtained via inverse symmetry or local linearity, then shows the subsequent matching step. This ordering will be highlighted to confirm that the classical formulas are consequences rather than inputs. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation presented as independent geometric foundation

full rationale

The paper extends a prior geometric construction from polynomials to elementary functions by defining the derivative directly as the slope coefficient of a tangent line via inverse symmetry and local linear structure. The claimed logical sequence (Tangent, Local Linear Structure, Limit formalisation) treats the tangent as a primitive geometric object from which differentiation rules are derived, with consistency to standard analysis shown afterward rather than presupposed. No quoted step reduces a result to a fitted parameter, self-citation chain, or renamed input by construction; the framework is self-contained against external benchmarks once the geometric primitives are accepted. This is the most common honest outcome for foundational reformulations that do not invoke uniqueness theorems or load-bearing self-citations for the target formulas.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on geometric intuitions as foundational axioms rather than introducing new free parameters or entities; the central claim rests on the assumption that tangent-line geometry can be defined prior to limits.

axioms (1)

domain assumption The tangent line can be interpreted geometrically to assign a slope coefficient at each point independently of the limit definition of the derivative
Invoked as the starting point for constructing derivatives of all listed elementary functions.

pith-pipeline@v0.9.0 · 5675 in / 1247 out tokens · 46330 ms · 2026-05-21T09:01:06.738426+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages · 2 internal anchors

[1]

Kapanadze, D. (2026). A limit-free algebraic–geometric construction of the derivative in the class of polynomial functions. arXiv preprint arXiv:2604.20888. Available at: https://arxiv.org/abs/2604.20888erences

work page internal anchor Pith review Pith/arXiv arXiv 2026
[2]

Cauchy, A.-L. (1821). Cours d’Analyse de l’École Royale Polytechnique. Paris

work page
[3]

Weierstrass, K. (1894). Mathematische Werke. Berlin

work page
[4]

Apostol, T. M. (1974). Mathematical Analysis (2nd ed.). Addison–Wesley

work page 1974
[5]

Rudin, W. (1976). Principles of Mathematical Analysis (3rd ed.). McGraw–Hill

work page 1976
[6]

The Derivative without Limits

Shisha, O. (1986). “The Derivative without Limits.” Journal of Mathematical Analysis and Applications, 118, 365–379

work page 1986
[7]

Derivatives without Limits

Sangwin, C. J. (2005). “Derivatives without Limits.” MSOR Connections, 5(2), 45–50

work page 2005
[8]

Zhang, X., & Tong, Y . (2018). A Calculus without Limits. arXiv preprint arXiv:1802.03029. Available at: https://arxiv.org/abs/2604.20888

work page internal anchor Pith review Pith/arXiv arXiv 2018
[9]

A Sensible Approach to the Calculus

Tall, D. (2010). “A Sensible Approach to the Calculus.” Paper presented at the National and International Meeting on the Teaching of Calculus, Pueblo, Mexico

work page 2010
[10]

Robinson, A. (1966). Non - standard Analysis . North-Holland

work page 1966
[11]

Keisler, H. J. (1986). Elementary Calculus : An Infinitesimal Approach . Prindle, Weber & Schmidt

work page 1986
[12]

Kapanadze, D. (2000). Foundations of Mathematical An alysis and the School Mathematics Course ( A Historical – Methodological Review ). Physics and Mathematics at School , No. 116. Available online at Academia.edu. (in Georgian)

work page 2000

[1] [1]

Kapanadze, D. (2026). A limit-free algebraic–geometric construction of the derivative in the class of polynomial functions. arXiv preprint arXiv:2604.20888. Available at: https://arxiv.org/abs/2604.20888erences

work page internal anchor Pith review Pith/arXiv arXiv 2026

[2] [2]

Cauchy, A.-L. (1821). Cours d’Analyse de l’École Royale Polytechnique. Paris

work page

[3] [3]

Weierstrass, K. (1894). Mathematische Werke. Berlin

work page

[4] [4]

Apostol, T. M. (1974). Mathematical Analysis (2nd ed.). Addison–Wesley

work page 1974

[5] [5]

Rudin, W. (1976). Principles of Mathematical Analysis (3rd ed.). McGraw–Hill

work page 1976

[6] [6]

The Derivative without Limits

Shisha, O. (1986). “The Derivative without Limits.” Journal of Mathematical Analysis and Applications, 118, 365–379

work page 1986

[7] [7]

Derivatives without Limits

Sangwin, C. J. (2005). “Derivatives without Limits.” MSOR Connections, 5(2), 45–50

work page 2005

[8] [8]

Zhang, X., & Tong, Y . (2018). A Calculus without Limits. arXiv preprint arXiv:1802.03029. Available at: https://arxiv.org/abs/2604.20888

work page internal anchor Pith review Pith/arXiv arXiv 2018

[9] [9]

A Sensible Approach to the Calculus

Tall, D. (2010). “A Sensible Approach to the Calculus.” Paper presented at the National and International Meeting on the Teaching of Calculus, Pueblo, Mexico

work page 2010

[10] [10]

Robinson, A. (1966). Non - standard Analysis . North-Holland

work page 1966

[11] [11]

Keisler, H. J. (1986). Elementary Calculus : An Infinitesimal Approach . Prindle, Weber & Schmidt

work page 1986

[12] [12]

Kapanadze, D. (2000). Foundations of Mathematical An alysis and the School Mathematics Course ( A Historical – Methodological Review ). Physics and Mathematics at School , No. 116. Available online at Academia.edu. (in Georgian)

work page 2000