Constructive Differential Calculus An Algebraic Formulation of Leibnizian Calculus

1. Introduction

Historically, Leibniz described differentials ($dx,dy$) as “incomparably small” quantities. Bishop Berkeley famously mocked them as the “ghosts of departed quantities" because they seemed non-zero during the calculation but magically became zero at the end. Intuitively, an infinitesimal can be conceptualized as an infinite sequence of real numbers definitively smaller, in absolute value, than any real number. This means that for any positive real number, after a certain position in the sequence, all subsequent elements are smaller than the chosen number. While this is a highly natural definition of an infinitesimal, it implies that the resulting class of infinitesimals forms an algebraic ring rather than an algebraic field. This limitation is fundamentally due to the lack of the product annihilation property; namely, two nonzero infinitesimals can multiply to yield a null sequence. Consider, for instance, the following regular sequences:

$$a_n=\left(1,0,{\displaystyle \frac{1}{3}},0,{\displaystyle \frac{1}{5}},0,\dots \right)\to 0$$

$$b_n=\left(0,{\displaystyle \frac{1}{2}},0,{\displaystyle \frac{1}{4}},0,{\displaystyle \frac{1}{6}},\dots \right)\to 0$$

$$a_n·b_n=(0,0,0,0,0,0,\dots )$$

Numerous foundational frameworks have been elaborated to resolve this inadequacy by providing a definition of infinitesimals that forces them into an algebraic field where the product annihilation property strictly holds—meaning a product of infinitesimals yields a null sequence if and only if at least one of the factors is itself a null sequence. However, these classical solutions require sophisticated logical and algebraic machinery [22,23,27], such as model theory, free ultrafilters, and the Axiom of Choice, resulting in heavily non-constructive axiomatic systems [3,12,18,20,21].

In the 19th century, Weierstrass and Cauchy replaced infinitesimals with the topology of limits ($ϵ-\delta$), introducing a rigor that nevertheless rigidified the fluidity of calculus, separating the praxis of physicists (who treat differentials as fractions and real increments) from the formal rigor of mathematicians [6].

In 1960, Abraham Robinson resolved the foundational crisis by establishing Non-Standard Analysis (NSA) through the extension of the real numbers to the field of hyperreals (*R). However, classical NSA relies heavily on model theory and non-constructive topological objects such as free ultrafilters, whose existence depends on the Axiom of Choice. For educational and applied physics applications, free ultrafilters represent an overkill, introducing a heavy “logical cannon” to solve purely algebraic problems.

In this paper, we develop an intuition arising in a study of Archimedes’ infinitesimals [19], and propose a constructive solution that captures the true underlying spirit of the Leibnizian intuition of the differential. We assume that infinitesimals are sequences that, in absolute value, are definitely smaller than any real number, and are either definitely zero or definitely different from zero (in general, a property holds definitely for a sequence if it holds universally after a certain index). However, within such a broad definition, it is usually difficult to decide a priori whether a generic sequence satisfies these conditions. To avoid relying on non-constructive tools analogous to ultrafilters to force decidability, we adopt a minimalist, constructive approach.

The core idea is straightforward. We initiate the calculus from simple, foundational sequences. Let x be a variable on real numbers; then any x-dependent infinitesimal sequence that is always different from zero after some element is a basic differential. When we apply term-by-term standard algebraic operations to basic differentials, we obtain definitely nonzero infinitesimals (see Theorem 2). Moreover, any differential $df\left(x\right)$ of a function f in the class of differentiable functions is definitely a nonzero differential, according to the Taylor-McLaurin expansion (see Theorem 7).

Crucially, these constructive infinitesimals are precisely the tools required to develop differential calculus rigorously. The “ghost” is thus endowed with a solid physical body: it is simply a well-behaved, regular sequence.

In standard calculus textbooks, students are told that they can discard ${\left(dx\right)}^2$ because it “goes to zero faster than $dx$.” To a beginner, this often feels like an illegal sleight of hand. Within our framework, if differentials are treated as polynomial structures in the generator $dx$, “neglecting” higher-order terms ceases to be a heuristic shortcut. It becomes a rigorous algebraic operation of truncation or extraction of the principal part. Consider the classic example of squaring a differential step:

$$d\left(x^2\right)=2x·\left\{{\displaystyle \frac{2x}{n}}\right\}+{\left\{{\displaystyle \frac{1}{n}}\right\}}^2={\left\{{\displaystyle \frac{2x}{n}}+{\displaystyle \frac{1}{n^2}}\right\}}_{n\in \mathbb{N}}$$

This expression is a polynomial structure. When seeking the macroscopic differential, we are simply applying a projection operator—the “Leibniz Shadow” (Sh)—that isolates the lowest-degree non-zero monomial with respect to $dx$:

$$\mathrm{Sh}\left(\left\{{\displaystyle \frac{2x}{n}}+{\displaystyle \frac{1}{n^2}}\right\}\right)=\mathrm{Sh}\left(2xdx+{\left(dx\right)}^2\right)=2xdx$$

There is no mysticism, no limits, and no ambiguous dropping of terms. One is merely sorting a polynomial structure by its lowest power.

1.1. The Algebraic Big-O Notation

To formally manage asymptotic remainders without invoking dynamic limits, we define the $\mathcal{O}$ notation within the sequential ring.

Definition 1 

($\mathcal{O}$ Notation). Let $\alpha \left(dx\right)$ and $\beta \left(dx\right)$ be two differentials. We write $\alpha \left(dx\right)=\mathcal{O}\left(\beta \right(dx\left)\right)$ if there exists a real constant $M>0$ and an index $N\in \mathbb{N}$ such that for all $n>N$:

$$\left|{\displaystyle \frac{\alpha \left(dx\right)}{\beta \left(dx\right)}}\right|\le M$$

(1)

In particular, $\mathcal{O}\left(d{x}^k\right)$ denotes a sequence whose lowest-degree monomial with respect to $dx$ has an exponent $\ge k$, ensuring that $\mathrm{Sh}\left({\displaystyle \frac{\mathcal{O}\left(d{x}^k\right)}{d{x}^{k-1}}}\right)=0$. while $\beta \left(x\right)$ is an $o\left(\alpha \right(x\left)\right)$, that is, $\beta \left(x\right)$ is a “small o” of $\alpha \left(x\right)$, if $\beta \left(x\right)/\alpha \left(x\right)$ is an infinitesimal.

The $\mathcal{O}$ notation is due to [2,16]. Donald Knuth addressed its importance in computer science and a constructive notion of calculus [13,14] using approaches similar to the present one. Here, a more radical vision is introduced, which is based on the notion of nonzero definiteness. The crucial role of $\mathcal{O}$ notation in the theory of numbers was demonstrated by Helge von Koch [15], who formulated the Riemann Hypothesis in terms of $\mathcal{O}$ notation.

Remark 1 

(The Dual Worlds of Big-O: From Infinitesimals to Infinities). The operational calculus of the CDC utilizes the $\mathcal{O}$-notation as an intrinsic scaling metric. It is worth noting that while this work primarily focuses on the local microscopic domain, where the seed $dx$ vanishes. However, the algebraic rules of Landau’s symbols encapsulate a flawless structural duality between infinitesimals and infinities via the inversion mapping $\omega ={\displaystyle \frac{1}{dx}}$.

Under this transformation, the hierarchical inversion reshapes the behavior of the scale envelopes. In the infinitesimal realm, higher exponents imply tighter convergence containment, meaning $\mathcal{O}\left(d{x}^b\right)\subset \mathcal{O}\left(d{x}^a\right)$ for $b>a$ (in this case $\mathcal{O}\left(\alpha \right)$ is the set of infinitesimals that are an $\mathcal{O}\left(\alpha \right)$). Conversely, in the transfinite realm of infinities, higher exponents dictate faster growth divergence, yielding the inverted inclusion $O\left({\omega }^a\right)\subset O\left({\omega }^b\right)$.

Crucially, the underlying multiplicative algebra remains invariant: $O\left(x^a\right)·O\left(x^b\right)=O\left(x^{a+b}\right)$ holds uniformly in both environments. This symmetry highlights that the CDC framework does not merely patch local calculus; by organizing the computable asymptotic scales, it establishes a unified algebraic language capable of traversing the polar extremes of mathematical analysis—from Leibniz’s localized differentials to the global boundaries of asymptotic expansions.

1.2. The Differential Formulation of Limits

A major pedagogical and foundational advantage of the CDC framework is its capacity to fully internalize the notion of a limit, translating the classical static $ϵ$-$\delta$ quantification into a direct, algebraic statement using constructive differentials. The following theorem is an easy consequence of the notion of limit and of an infinitesimal sequence.

Theorem 1 

(Limits in CDC). Let $f:E\to \mathbb{R}$ be a real-valued function defined on a subset $E\subset \mathbb{R}$.

$$\underset{x\to x_0}{lim}f\left(x\right)=L\iff \mathrm{Sh}\left(f(x_0+dx)\right)=L.$$

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2. The Constructive Differential Calculus (CDC)

To bypass the non-constructive limitations of NSA and preserve Leibniz’s intuitive clarity, we introduce the Constructive Differential Calculus (CDC). By grounding the differential operator on standard infinitesimal sequences, the “aut-aut” property (every infinitesimal is either eventually zero or eventually non-zero) becomes an intrinsic analytic property, completely dissolving zero-divisors without external constraints.

2.1. The Minimalist Axiomatic System of CDC

Let ${\mathcal{D}}^{+}$ be the algebraic extension of the real numbers satisfying the following core postulates:

Axiom 1 

(The Differential Seed). The differential of the independent variable x, for $x\ne 0$, is a definitely nonzero x-infinitesimal defined by the variable-dependent sequence (${\mathbb{N}}^{+}=\mathbb{N}-\left\{0\right\}$):

$$dx={\left\{d{x}_n\right\}}_{n\in {\mathbb{N}}^{+}}={\{1/n+e(x,n)\}}_{n>0}$$

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where, for any value of x in real numbers, $e(x,n)$ is either definitely equal to zero: $e(x,n)=0$ for n greater than some value, or it is $o(1/n)$. In this way, the dependence of $d\left(x\right)$ on x is evident, but the exact value of x does not affect the results of the operations that we apply to the expressions including $dx$. In many cases, we shortly consider $dx=\{1/n\}$.

Axiom 2 

(Differential of a Function). The differential operator d applied to a function $f\left(x\right)$ is defined as the increment of $f\left(x\right)$ on the microscopic step $dx$:

$$df\left(x\right)\equiv f(x+dx)-f\left(x\right)$$

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Axiom 3 

(Differentiability and First Derivative). A function $f\left(x\right)$ is defined to be differentiable at x if its generated differential $df\left(x\right)$ can be structurally decomposed as a linear function of $dx$ up to a higher-order remainder:

$$df\left(x\right)=Df\left(x\right)dx+\mathcal{O}\left(d{x}^2\right)$$

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where $Df\left(x\right)$ is a value, shortly denoted also by $f^{\prime }\left(x\right)$, which is a real coefficient called the first derivative of f in x.

Axiom 4 

(Notational Priority). The differentiation operator d takes precedence over powers when applied to the differentiated variable:

$$d{x}^n\equiv {\left(dx\right)}^n$$

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Axiom 5 

(Order Zero Derivative). The differential operator of order zero yields the original identical function:

$$D^{0}f\left(x\right)=f\left(x\right)$$

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Axiom 6 

(Higher-order Derivatives). Higher-order derivative of order $n+1$ ($n\in \mathbb{N}$), when it exists, is the derivative of the n order derivative:

$$D^{n+1}f\left(x\right)=D\left(D^{n}f\left(x\right)\right)$$

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Axiom 7 

(The Differential of a Differentiable Function). When $f\left(x\right)$ is differentiable, then its differential can be decomposed as:

$$df\left(x\right)=(f^{\prime }\left(x\right)+d{f}^{\prime }\left(x\right))dx$$

(9)

Axiom 8 

(The Shadow Operator or Macroscopic Projection). The derivative $f^{\prime }\left(x\right)$ is defined as the projection operator Sh (Shadow) that extracts the constant term (degree zero) in a polynomial $P\left(dx\right)$ in $dx$. For example:

$$\mathrm{Sh}\left(a+bdx+cd{x}^2+\dots \right)=a$$

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In particular, $Sh$ extracts the derivative of $f\left(x\right)$ in the ratio ${\displaystyle \frac{df\left(x\right)}{dx}}$:

$$f^{\prime }\left(x\right)\equiv \mathrm{Sh}\left({\displaystyle \frac{df\left(x\right)}{dx}}\right)$$

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We define $d^{2}f\left(x\right)$:

$$d^{2}f\left(x\right)=d{f}^{\prime }\left(x\right)=f^{\prime \prime }\left(x\right)d{x}^2+\mathcal{O}\left(d{x}^3\right)$$

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This axiom, together with the previous axiom, implies:

$$\begin{array}{cc}\hfill & df\left(x\right)=f^{\prime }\left(x\right)dx+\left(f^{\prime \prime }\left(x\right)dx+\mathcal{O}\left(d{x}^2\right)\right)dx=\hfill \\ \hfill & f^{\prime }\left(x\right)dx+f^{\prime \prime }\left(x\right)d{x}^2+\mathcal{O}\left(d{x}^3\right)=\hfill \\ \hfill & f^{\prime }\left(x\right)dx+{\displaystyle \frac{d^{2}f\left(x\right)}{d{x}^2}}d{x}^2+\mathcal{O}\left(d{x}^3\right)\hfill \end{array}$$

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That motivates Leibniz’s notation for the second derivative, and in general, ${\displaystyle \frac{d^{n}f\left(x\right)}{d{x}^n}}$ for the n-order derivative. Namely:

$${\displaystyle \frac{d^{2}f\left(x\right)}{d{x}^2}}=f^{\prime \prime }\left(x\right)+\mathcal{O}\left(dx\right)$$

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And:

$$df\left(x\right)=\left(f^{\prime }\left(x\right)+d^{2}f\left(x\right)\right)dx$$

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Shadow operator (Sh) is a constructive counterpart to the standard part mapping originally introduced within Nonstandard Analysis by Robinson [23] and geometrically framed as the “shadow” of a nonstandard number by Nelson [20].

2.2. Algebraic Derivation Example

Consider the function $f\left(x\right)=x^2$. Applying Axioms 2 and 3, we compute the microscopic increment:

$$df\left(x\right)={(x+dx)}^2-x^2=2xdx+d{x}^2$$

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Substituting the explicit definition of Axiom 1, this evaluates term-by-term to:

$$df\left(x\right)=2x\left\{d{x}_n\right\}+{\left\{d{x}_n\right\}}^2={\left\{2xd{x}_n+d{x}_n^2\right\}}_{n\in \mathbb{N}}$$

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Since $dx{\ne }_{def}0$ (definitely nonzero), dividing by $dx$ is a valid algebraic operation:

$${\displaystyle \frac{df\left(x\right)}{dx}}={\displaystyle \frac{2xdx+{\left(dx\right)}^2}{dx}}=2x+dx$$

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Applying Axiom 8 (Shadow Operator), we isolate the macroscopic component by dropping the additive differentials:

$$f^{\prime }\left(x\right)=\mathrm{Sh}(2x+dx)=2x$$

(19)

The derivative is obtained via elementary algebra, completely bypassing limit processes.

Remark 2. 

The derivative of a function $f\left(x\right)$ at a point $x_0$ is $f^{\prime }\left(x_0\right)$. When we write $f^{\prime }\left(x\right)$, we mean the derivative function giving the derivatives of $f\left(x\right)$ at x, where x varies in a given, specified (or intended) interval of variability of the variable x.

3. Constructive Differentials and their Asymptotic Ordering

Theorem 2 

(Algebraic Closure of Nonzero Definiteness). Let us assume that basic differentials $dx,dy,\dots$ are definitely nonzero infinitesimals, then:

i)

Summing or subtracting definitely distinct differentials provides a definitely nonzero infinitesimal;

ii)

Multiplying or dividing differentials provides a definitely nonzero infinitesimal;

iii)

Operations $c*dx$ between $dx$ and a real number c provide ${\{c*d{x}_n\}}_{n\in \mathbb{N}}$ (division of $dx$ by zero is not allowed).

Proof. 

Let $\alpha ={\left\{a_n\right\}}_{n\in \mathbb{N}}$ and $\beta ={\left\{b_n\right\}}_{n\in \mathbb{N}}$ be two definitely nonzero infinitesimal in ${\mathcal{D}}^{*}+$. By hypothesis, in $\alpha$ and $\beta$ there exist indices $N_a,N_b\in \mathbb{N}$ such that $a_n\ne 0$ for all $n>N_a$ and $b_n\ne 0$ for all $n>N_b$. Let $N_0=max(N_a,N_b)$.

• Summation ($\alpha \pm \beta$): If $\alpha$ and $\beta$ are definitely different, their pointwise sum or difference is surely definitely nonzero.
• Product ($\alpha ·\beta$): The multiplication is defined pointwise by the sequence ${\{a_n·b_n\}}_{n\in \mathbb{N}}$. For any $n>N_0$, both $a_n$ and $b_n$ are non-zero elements, therefore $a_n·b_n\ne 0$ for all $n>N_0$. Hence, the product is definitely nonzero.
• Quotient ($\alpha /\beta$): The division is defined pointwise by ${\{a_n/b_n\}}_{n\in \mathbb{N}}$ for $n>N_0$. Since $b_n\ne 0$ for all $n>N_0$, the algebraic operation is well-defined. The quotient of two non-zero real numbers is inherently non-zero in $\mathbb{R}$, ensuring $\alpha /\beta {\ne }_{def}0$.

□

Remark 3. 

In the following, the relations between infinitesimals (equality, ordering) are intended, even if not explicitly expressed, asdefinitely holdingrelations.

By abstracting this theorem from a specific numerical seed, we have successfully formulated a representation-independent algebraic property of “definitely nonzero” as an invariant structural trait of Constructive Differential Calculus. No matter what initial valid progression generates differentials, the operations of calculus will never accidentally degrade an active, non-zero infinitesimal fraction into a zero-divisor.

The algebraic closure established by Theorem 2 defines a space of core differentials ${\mathcal{D}}^{+}$ of definitely non-zero infinitesimals generated by the base seed $dx=\left\{d{x}_n\right\}$, that can be totally ordered. However, this ordering highlights a fundamental topological property of the CDC framework: the violation of the classical Archimedean property.

Proposition 1 

(Non-Archimedean Property). The space of constructive differentials ${\mathcal{D}}^{+}$ is non-Archimedean with respect to the standard real field $\mathbb{R}$. For any base seed $dx\in {\mathcal{D}}^{+}$ and any macroscopically large standard real number $r\in {\mathbb{R}}^{+}$, there exists no standard integer $n\in \mathbb{N}$ such that $n·dx>r$.

Proof. 

Let $n\in \mathbb{N}$ be any fixed, standard natural number. By applying the linear properties of the Shadow operator (Sh), which maps the infinitesimal domain back to the standard real continuum, we evaluate the macroscopic projection of the magnified differential:

$$\mathrm{Sh}(n·dx)=n·\mathrm{Sh}\left(dx\right)=n·0=0.$$

(20)

Since $\mathrm{Sh}\left(r\right)=r>0$ for any standard positive real number, it follows that $\mathrm{Sh}(n·dx)<\mathrm{Sh}\left(r\right)$, proving that $n·dx<r$ for all $n\in \mathbb{N}$. □

This structural insulation ensures that while the seed $dx$ remains actively non-zero and operationally dynamic within the microscopic layer, it is safely bounded beneath the standard real line, preventing any algebraic spillover into the macroscopic continuum before the application of the Shadow operator.

Remark 4 

(Nonzero Infinitesimal Identity and Distinction). Within the class of definitely non-zero infinitesimals, the concepts of identity and distinction are defined operationally through the lens of definiteness. Two infinitesimal sequences $\alpha =\left\{a_n\right\}$ and $\beta =\left\{b_n\right\}$ are considered equal if they are definitely equal, meaning:

$$\exists N\in \mathbb{N}:\forall n>N,a_n=b_n.$$

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Conversely, they are considered distinct if they are definitely distinct, meaning:

$$\exists N\in \mathbb{N}:\forall n>N,a_n\ne b_n.$$

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Scholium 1 

(Nonzero infinitesimals are partially ordered). It is crucial to observe that in the universe of definitely nonzero infinitesimals, there exist sequences that are neither definitely equal nor definitely distinct. Consider, for instance, the two sequences defined by:

$$\alpha =\left\{{\displaystyle \frac{1}{n}}\right\},\beta =\left\{{\displaystyle \frac{1}{n}}+{\displaystyle \frac{1+{(-1)}^n}{n^2}}\right\}.$$

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Both sequences are definitely non-zero infinitesimals; however, because of the oscillating factor ${(-1)}^n$, they coincide exactly for every odd n (hence they are not definitely distinct), while they diverge for every even n (hence they are not definitely equal).

Within the restricted class of definitely non-zero infinitesimals, the ordering is not total, and they can violate the Law of Excluded Middle, rendering any algebraic development of differential operations impossible without non-constructive logical or algebraic expedients.

However, in the subclass of constructive differentials, the ordering is total, as we will show in the following subsection.

The foundational power of the Constructive Differential Calculus (CDC) lies precisely in its screening mechanism, making such oscillatory hybrids mathematically inadmissible. Within the domain of constructive differentials, every sequence is strictly constrained to the binary reality of being either definitely equal or definitely distinct.

Following the algebraic closure established by Theorem 2, we formally define the restricted mathematical environment within which the operational calculus of the CDC takes place.

Definition 2 

(The Class ${\mathcal{D}}^{+}$ of Constructive Differentials). The class ${\mathcal{D}}^{+}$ of strictly positive constructive differentials consists of all vanishing sequences generated by the algebraic closure of definitely non-zero differentials $d^{n}f$, with $n\in \mathbb{N}$.

A profound consequence of this operational filter is the immediate, automated exclusion of pathologically oscillating variations.

Example 1 

(The Non-Admissibility of some differentials). Consider the differential of:

$$f\left(x\right)=x·sin\left({\displaystyle \frac{\pi }{2x}}\right)$$

When we apply the usual rules of differential calculus, we find that this differential at $x=0$ is:

$$d{f}_n=f\left({\displaystyle \frac{1}{n}}\right)={\displaystyle \frac{1}{n}}·sin\left({\displaystyle \frac{\pi }{2·\frac{1}{n}}}\right)={\displaystyle \frac{1}{n}}·sin\left({\displaystyle \frac{\pi }{2}}n\right)$$

giving:

for $n=1\Rightarrow d{f}_1=1·sin(\pi /2)=1$

for $n=2$$\Rightarrow d{f}_2={\displaystyle \frac{1}{2}}·sin\left(\pi \right)=\mathbf{0}$

for $n=3\Rightarrow d{f}_3={\displaystyle \frac{1}{3}}·sin(3\pi /2)=-{\displaystyle \frac{1}{3}}$

for $n=4$$\Rightarrow d{f}_4={\displaystyle \frac{1}{4}}·sin\left(2\pi \right)=\mathbf{0}$

That is, a sequence having infinite zeros:

$$\left\{1,\mathbf{0},-{\displaystyle \frac{1}{3}},\mathbf{0},{\displaystyle \frac{1}{5}},\mathbf{0},-{\displaystyle \frac{1}{7}},\mathbf{0},\dots \right\}$$

Failing the Nonzero Definiteness criterion, $d(x·sin\left({\displaystyle \frac{\pi }{2x}}\right))$ is structurally inadmissible within ${\mathcal{D}}^{+}$. The failure to be definitely nonzero is sufficient for its exclusion.

A total ordering can be defined on constructive differentials, which is based on the real exponents of $dx$.

Proposition 2 

(Differential Asymptotic Ordering). For any two constructive differentials expressed as real powers of the fundamental seed, $d{x}^a$ and $d{x}^b$ with $a,b\in {\mathbb{R}}^{+}$, the strict operational order is governed by the exponents:

$$d{x}^a<d{x}^b\iff a>b.$$

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Proof. 

Let $dx=\left\{d{x}_n\right\}$ be a strictly positive vanishing sequence ($0<d{x}_n<1$ definitively). For any $a>b$, the ratio ${\displaystyle \frac{d{x}_n^a}{d{x}_n^b}}=d{x}_n^{a-b}$ constitutes a vanishing sequence since $a-b>0$. Therefore, there exists an index $N\in \mathbb{N}$ such that for all $n>N$, $d{x}_n^{a-b}<1$, which implies $d{x}_n^a<d{x}_n^b$. □

This ensures that the hierarchy of scales within the non-Archimedean field $\mathcal{D}$ remains perfectly determined for any fractional or irrational differential variations generated by function composition, where 0 is considered equal to any definitely zero infinitesimal:

$$0<\dots <d{x}^3<d{x}^2<dx<d{x}^{1/2}<d{x}^{1/3}<\dots <r,\forall r\in {\mathbb{R}}^{+}.$$

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4. Fundamental Proofs of Calculus in CDC

4.1. Sum Derivative

Theorem 3 

(Sum Rule). Let $f\left(x\right)$ and $g\left(x\right)$ be two functions derivable in x.

$$D(f\left(x\right)+g\left(x\right))=f^{\prime }\left(x\right)+g^{\prime }\left(x\right)$$

Proof. 

The differential of $S\left(x\right)=f\left(x\right)+g\left(x\right)$ is obtained by Axiom 2:

$$dS\left(x\right)=S(x+dx)-S\left(x\right)=\left(f(x+dx)+g(x+dx)\right)-\left(f\left(x\right)+g\left(x\right)\right)$$

Rearranging terms:

$$dS\left(x\right)=\left(f(x+dx)-f\left(x\right)\right)+\left(g(x+dx)-g\left(x\right)\right)$$

By Axiom 6, substituting the structural decompositions:

$$dS\left(x\right)=\left(f^{\prime }\left(x\right)dx+\mathcal{O}\left(d{x}^2\right)\right)+\left(g^{\prime }\left(x\right)dx+\mathcal{O}\left(d{x}^2\right)\right)$$

giving:

$$dS\left(x\right)=\left(f^{\prime }\left(x\right)+g^{\prime }\left(x\right)\right)dx+\mathcal{O}\left(d{x}^2\right)$$

Applying the Shadow Operator (Axiom 8):

$$S^{\prime }\left(x\right)\equiv \mathrm{Sh}\left({\displaystyle \frac{dS\left(x\right)}{dx}}\right)=\mathrm{Sh}\left({\displaystyle \frac{\left(f^{\prime }\left(x\right)+g^{\prime }\left(x\right)\right)dx+\mathcal{O}\left(d{x}^2\right)}{dx}}\right)$$

Which simplifies to:

$$S^{\prime }\left(x\right)=\mathrm{Sh}\left(f^{\prime }\left(x\right)+g^{\prime }\left(x\right)+\mathcal{O}\left(dx\right)\right)$$

Because $\mathrm{Sh}\left(\mathcal{O}\right(dx\left)\right)=0$, we obtain:

$$S^{\prime }\left(x\right)=f^{\prime }\left(x\right)+g^{\prime }\left(x\right)$$

□

4.2. The Product Rule

Theorem 4 

(Product Rule). Let $f\left(x\right)$ and $g\left(x\right)$ be regular functions. Then:

$$\mathrm{Sh}\left({\displaystyle \frac{d\left(f\right(x\left)g\right(x\left)\right)}{dx}}\right)=f^{\prime }\left(x\right)g\left(x\right)+f\left(x\right)g^{\prime }\left(x\right)$$

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Proof. 

By Axiom 2, we have:

$$\begin{array}{cc}\hfill d\left(f\right(x\left)g\right(x\left)\right)& =f(x+dx)g(x+dx)-f\left(x\right)g\left(x\right)\hfill \\ \hfill & =\left(f\left(x\right)+df\left(x\right)\right)\left(g\left(x\right)+dg\left(x\right)\right)-f\left(x\right)g\left(x\right)\hfill \\ \hfill & =f\left(x\right)g\left(x\right)+f\left(x\right)dg\left(x\right)+g\left(x\right)df\left(x\right)+df\left(x\right)dg\left(x\right)-f\left(x\right)g\left(x\right)\hfill \\ \hfill & =f\left(x\right)dg\left(x\right)+g\left(x\right)df\left(x\right)+df\left(x\right)dg\left(x\right)\hfill \end{array}$$

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Dividing by $dx$ yields:

$${\displaystyle \frac{d\left(f\right(x\left)g\right(x\left)\right)}{dx}}=f\left(x\right){\displaystyle \frac{dg\left(x\right)}{dx}}+g\left(x\right){\displaystyle \frac{df\left(x\right)}{dx}}+{\displaystyle \frac{df\left(x\right)dg\left(x\right)}{dx}}$$

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Since $df\left(x\right)dg\left(x\right)=\mathcal{O}\left(d{x}^2\right)$ by Axiom 6, the last term reduces to $\mathcal{O}\left(dx\right)$. Applying the Shadow projection (Axiom 8) isolates the constant terms, giving $f\left(x\right)g^{\prime }\left(x\right)+g\left(x\right)f^{\prime }\left(x\right)$. □

4.3. Power Derivatives

Theorem 5 

(Power Rule). Let $f\left(x\right)=x^n$ where $n\in \mathbb{N}$. Then $d\left(x^n\right)=\left(n{x}^{n-1}\right)dx+\mathcal{O}\left(d{x}^2\right)$.

Proof. 

By applying Axiom 2, the operational differential of the power function is given by:

$$d\left(x^n\right)={(x+dx)}^n-x^n$$

(29)

Expanding the term ${(x+dx)}^n$ using the exact algebraic Binomial Theorem yields:

$$d\left(x^n\right)=\left(x^n+\left({\displaystyle \genfrac{}{}{0pt}{}{n}{1}}\right)x^{n-1}dx+\left({\displaystyle \genfrac{}{}{0pt}{}{n}{2}}\right)x^{n-2}d{x}^2+\dots +d{x}^n\right)-x^n$$

(30)

Canceling the $x^n$ terms and substituting the explicit binomial coefficients, we obtain:

$$d\left(x^n\right)=n{x}^{n-1}dx+{\displaystyle \frac{n(n-1)}{2}}{x}^{n-2}d{x}^2+\dots +d{x}^n$$

(31)

We can group all terms containing powers of $dx$ greater than or equal to 2 under the algebraic remainder definition:

$$d\left(x^n\right)=\left(n{x}^{n-1}\right)dx+\mathcal{O}\left(d{x}^2\right)$$

(32)

According to the structural differentiability condition established in Axiom 6, the macroscopic coefficient of the linear $dx$ term uniquely defines the derivative. Thus, the derivative is structurally identified as $n{x}^{n-1}$. □

4.4. The Chain Rule

The extension of the asymptotic ordering to the real exponent space ${\mathbb{R}}^{+}$ allows for a non-epsilon-delta proof of the chain rule, via pure structural algebra.

Theorem 6 

(The Chain Rule). Let f be differentiable at a point $x_0$, and let g be differentiable at $y_0=f\left(x_0\right)$. Then the composite function $h\left(x\right)=(g\circ f)\left(x\right)$ is differentiable at $x_0$, and its derivative satisfies:

$$h^{\prime }\left(x_0\right)=g^{\prime }\left(f\left(x_0\right)\right)·f^{\prime }\left(x_0\right).$$

(33)

Proof. 

Let $dx\in {\mathcal{D}}^{+}$ be the fundamental microscopic seed. We evaluate the total variation of the internal function f around $x_0$. By definition, the internal algebraic increment $df$ is expressed as:

$$df=f(x_0+dx)-f\left(x_0\right)=f^{\prime }\left(x_0\right)dx+O\left(d{x}^{\alpha }\right),\mathrm{where}\alpha >1.$$

(34)

Next, we evaluate the composition $h(x_0+dx)=g\left(f(x_0+dx)\right)=g(f\left(x_0\right)+df)$. Expanding the external function g algebraically around its standard base point $y_0=f\left(x_0\right)$ yields:

$$g(f\left(x_0\right)+df)=g\left(f\left(x_0\right)\right)+g^{\prime }\left(f\left(x_0\right)\right)df+O\left(d{f}^{\beta }\right),\mathrm{where}\beta >1.$$

(35)

Isolizing the total differential $dh=h(x_0+dx)-h\left(x_0\right)$ and substituting the explicit expression for $df$, we obtain the nested expansion:

$$dh=g^{\prime }\left(f\left(x_0\right)\right)\left[f^{\prime }\left(x_0\right)dx+O\left(d{x}^{\alpha }\right)\right]+O\left({\left[f^{\prime }\left(x_0\right)dx+O\left(d{x}^{\alpha }\right)\right]}^{\beta }\right).$$

(36)

Distributing the linear terms and invoking the algebraic properties of the scale envelopes, we can collect all components with exponents strictly greater than 1 into a single structural remainder:

$$dh=g^{\prime }\left(f\left(x_0\right)\right)f^{\prime }\left(x_0\right)dx+O\left(d{x}^{\gamma }\right)$$

(37)

where $\gamma =min(\alpha ,\beta )>1$. Since $\gamma >1$, the subset inclusion rule established previously guarantees that $O\left(d{x}^{\gamma }\right)\subset O\left(d{x}^1\right)$. To extract the macroscopic derivative, we divide the total variation by the linear seed $dx$:

$${\displaystyle \frac{dh}{dx}}=g^{\prime }\left(f\left(x_0\right)\right)f^{\prime }\left(x_0\right)+O\left(d{x}^{\gamma -1}\right).$$

(38)

Because $\gamma -1>0$, the remainder $O\left(d{x}^{\gamma -1}\right)$ is a pure infinitesimal sequence belonging to ${\mathcal{D}}^{+}$. Applying the linear Shadow operator (Sh) to project the microscopic ratio onto the standard real continuum yields:

$$h^{\prime }\left(x_0\right)=\mathrm{Sh}\left({\displaystyle \frac{dh}{dx}}\right)=\mathrm{Sh}\left(g^{\prime }\left(f\left(x_0\right)\right)f^{\prime }\left(x_0\right)+O\left(d{x}^{\gamma -1}\right)\right)=g^{\prime }\left(f\left(x_0\right)\right)·f^{\prime }\left(x_0\right).$$

(39)

This completes the proof. □

4.5. Taylor-McLaurin Expansion

In the CDC framework, this famous theorem is a direct consequence of Axioms 5, 6, and 7.

Theorem 7 

(Taylor-Maclaurin Expansion). Let $f\left(x\right)$ be an m-times structurally differentiable function. Then its microscopic increment can be exactly mapped by a polynomial expansion up to a higher-order remainder:

$$f(x+dx)=f\left(x\right)+f^{\prime }\left(x\right)dx+{\displaystyle \frac{D^{2}f\left(x\right)}{2!}}d{x}^2+\dots +{\displaystyle \frac{D^{m}f\left(x\right)}{m!}}d{x}^m+\mathcal{O}\left(d{x}^{m+1}\right)$$

(40)

Proof. 

By applying Axioms 2, 5, and 7 directly, the native differential expansion of $df\left(x\right)$ yields a linear combination of dominant monomials:

$$f(x+dx)=f\left(x\right)+D^{1}f\left(x\right)dx+D^{2}f\left(x\right)d{x}^2+\dots +D^{m}f\left(x\right)d{x}^m+\mathcal{O}\left(d{x}^{m+1}\right)$$

(41)

To contextualize this structure within classical power expansions, let us define a control polynomial $T\left(z\right)$ of degree m in the variable z:

$$T\left(z\right)\equiv f\left(x\right)+f^{\prime }\left(x\right)(z-x)+{\displaystyle \frac{D^{2}f\left(x\right)}{2!}}{(z-x)}^2+\dots +{\displaystyle \frac{D^{m}f\left(x\right)}{m!}}{(z-x)}^m$$

(42)

Evaluating this control polynomial at $z=x+dx$ yields:

$$T(x+dx)=f\left(x\right)+f^{\prime }\left(x\right)dx+{\displaystyle \frac{D^{2}f\left(x\right)}{2!}}d{x}^2+\dots +{\displaystyle \frac{D^{m}f\left(x\right)}{m!}}d{x}^m$$

(43)

Let us now compute the higher-order structural derivatives of $T\left(x\right)$ at x using Axiom 7. Due to the exact algebraic properties of monomial differentiation under the differential operator, the exponent falling chain $k(k-1)\dots (k-n+1)$ perfectly cancels out the $k!$ denominators. Thus, for any order $n\le m$, we obtain:

$$D^{n}T\left(x\right)=D^{n}f\left(x\right)$$

(44)

Since the actual function $f\left(x\right)$ and the control polynomial $T\left(x\right)$ share the same structural derivative coefficients up to order m within the sequential ring, their algebraic expansions evaluated on the increment $dx$ must match identically on all terms of degree $\le m$. Consequently, the true function evaluation $f(x+dx)$ and the polynomial translation $T(x+dx)$ can only differ starting from the next available algebraic rank, bounded by $\mathcal{O}\left(d{x}^{m+1}\right)$. This directly establishes:

$$f(x+dx)=T(x+dx)+\mathcal{O}\left(d{x}^{m+1}\right)$$

(45)

which produces the Taylor-Maclaurin expansion theorem exactly, completing the proof through the structural uniqueness of the control polynomial. □

Corollary 1 

(Nonzero Definiteness of a Differential at a Point). Let $f\left(x\right)$ be a non-constant, differentiable function in a neighborhood of a fixed real point $x_0\in \mathbb{R}$. If $f\left(x\right)$ admits a local Taylor expansion at $x_0$, then its differential at that point, $df\left(x_0\right)$, is definitely a nonzero infinitesimal.

Proof. 

By the properties of the Taylor expansion at a fixed point $x_0$, the local variation is governed by the first non-vanishing derivative at that point, such that $f^{\left(n\right)}\left(x_0\right)=c\in \mathbb{R}\setminus \left\{0\right\}$ for some finite $n\ge 1$. The operational core differential extracts this leading term:

$$df\left(x_0\right)={\displaystyle \frac{f^{\left(n\right)}\left(x_0\right)}{n!}}·d{x}^n=c·d{x}^n$$

The thesis follows directly from properties (ii) and (iii) of Theorem 2 (Algebraic Closure) that the sequence representing $df\left(x_0\right)$ is strictly and definitely nonzero. □

4.6. Trigonometric and Exponential Derivatives

Theorem 8. 

The derivative of $sinx$ is $cosx$.

Proof. 

We evaluate the infinitesimal increment of the sine function. Expanding via trigonometric identities through Axiom 2, we have:

$$d(sinx)=sin(x+dx)-sinx=sinx(cos(dx)-1)+cosxsin\left(dx\right)$$

(46)

To bound the transcendental functions of the infinitesimal increment $dx$, we consider the classical geometric properties of the unit circle. For a sufficiently small positive increment $dx$, the standard geometric inequality holds:

$$sin\left(dx\right)<dx<tan\left(dx\right)$$

(47)

Dividing by $sin\left(dx\right)$ yields $1<{\displaystyle \frac{dx}{sin\left(dx\right)}}<{\displaystyle \frac{1}{cos\left(dx\right)}}$. Since $cos\left(dx\right)<cos\left(0\right)=1$:

$${\displaystyle \frac{sin\left(dx\right)}{dx}}=1+\mathcal{O}\left(dx\right)\Rightarrow sin\left(dx\right)=dx+\mathcal{O}\left(d{x}^2\right)$$

(48)

Furthermore, using the identity $cos\left(dx\right)-1=-2{sin}^2(dx/2)$ (which derives from $cos\left(2\theta \right)={cos}^2\theta -{sin}^2\theta$), it follows that the cosine deviation is an infinitesimal of higher order:

$$cos\left(dx\right)-1=\mathcal{O}\left(d{x}^2\right)$$

(49)

We now construct the derivative ratio by dividing $d(sinx)$ by $dx$:

$${\displaystyle \frac{d(sinx)}{dx}}=sinx{\displaystyle \frac{cos\left(dx\right)-1}{dx}}+cosx{\displaystyle \frac{sin\left(dx\right)}{dx}}$$

(50)

Substituting the previous $\mathcal{O}$ evaluations of $sin\left(dx\right)$ and $cos\left(dx\right)-1$ into the ratio gives:

$${\displaystyle \frac{d(sinx)}{dx}}=sinx{\displaystyle \frac{\mathcal{O}\left(d{x}^2\right)}{dx}}+cosx(1+\mathcal{O}\left(dx\right))$$

(51)

which simplifies algebraically to:

$${\displaystyle \frac{d(sinx)}{dx}}=cosx+\mathcal{O}\left(dx\right)$$

(52)

Applying Axiom 8:

$$Dsin\left(x\right)=cos\left(x\right)$$

(53)

□

Theorem 9. 

The derivative of $cosx$ is $-sinx$.

Proof. 

We evaluate, through Axiom 2, the infinitesimal increment of the cosine function. Expanding the total differential via trigonometric identities yields:

$$d(cosx)=cos(x+dx)-cosx=cosx(cos(dx)-1)-sinxsin\left(dx\right)$$

(54)

To compute the derivative ratio, we divide the relation by the infinitesimal increment $dx$:

$${\displaystyle \frac{d(cosx)}{dx}}=cosx{\displaystyle \frac{cos\left(dx\right)-1}{dx}}-sinx{\displaystyle \frac{sin\left(dx\right)}{dx}}$$

(55)

Utilizing the trigonometric identity $cos\left(dx\right)-1=-2{sin}^2(dx/2)$, this deviation constitutes a higher-order infinitesimal:

$$cos\left(dx\right)-1=\mathcal{O}\left(d{x}^2\right)$$

(56)

Simultaneously, the first-order asymptotic expansion for the sine function gives ${\displaystyle \frac{sin\left(dx\right)}{dx}}=1+\mathcal{O}\left(dx\right)$. Substituting these relations back into the previous equation leads to:

$${\displaystyle \frac{d(cosx)}{dx}}=cosx{\displaystyle \frac{\mathcal{O}\left({\left(dx\right)}^2\right)}{dx}}-sinx(1+\mathcal{O}\left(dx\right))$$

(57)

Simplifying the algebraic orders of magnitude, we obtain:

$${\displaystyle \frac{d(cosx)}{dx}}=cosx·\mathcal{O}\left(dx\right)-sinx-sinx·\mathcal{O}\left(dx\right)$$

(58)

Since $cosx$ and $sinx$ are bounded functions on the real field, their product with an infinitesimal remains infinitesimal. Thus, the expression collapses to:

$${\displaystyle \frac{d(cosx)}{dx}}=-sinx+\mathcal{O}\left(dx\right)$$

(59)

Applying Axiom 8:

$$Dcos\left(x\right)=-sin\left(x\right)$$

(60)

□

Theorem 10. 

The derivative of $e^x$ is $e^x$.

Proof. 

The operational difference (Axiom 2) yields: $d\left(e^x\right)=e^{x+dx}-e^x=e^x(e^{dx}-1)$. Expanding the exponential function into its analytic series: $e^{dx}-1=dx+\mathcal{O}\left(d{x}^2\right)$. Dividing by $dx$, we obtain ${\displaystyle \frac{d\left(e^x\right)}{dx}}=e^x(1+\mathcal{O}\left(dx\right))$. The Shadow operator (Axiom 8) projects this to $e^x·1=e^x$. □

4.7. Logarithmic Derivatives

Theorem 11 

(Derivative of the Natural Logarithm). Let $f\left(x\right)=lnx$ defined for $x>0$. Then $d(lnx)=\left({\displaystyle \frac{1}{x}}\right)dx+\mathcal{O}\left(d{x}^2\right)$.

Proof. 

Let $y=e^x$, then $x=lny$. By Axiom 2 and expansion:

$$dy\equiv d{e}^x=e^{x}dx+\mathcal{O}\left(d{x}^2\right)$$

The differential $d(lny)$ is:

$$ln(y+dy)-lny$$

Replacing $y=e^x$ and $y+dy=e^{x+dx}$ via function inversion:

$$d(lny)=ln\left(e^{x+dx}\right)-ln\left(e^x\right)=(x+dx)-x=dx$$

Therefore:

$$d(lny)=dx$$

Now, we express $dx$ in terms of $dy$. Since $e^x\ne 0$ for any real number x:

$$dy=dx\left(e^x+{\displaystyle \frac{\mathcal{O}\left(d{x}^2\right)}{dx}}\right)=dx\left(e^x+\eta \right)$$

where $\eta \in \mathcal{O}\left(dx\right)$, and:

$$dx={\displaystyle \frac{dy}{e^x+\eta }}={\displaystyle \frac{1}{e^x\left(1+\frac{\eta }{e^x}\right)}}dy$$

Using a geometric series:

$$dx={\displaystyle \frac{1}{e^x}}\left(1-{\displaystyle \frac{\eta }{e^x}}+\dots \right)dy={\displaystyle \frac{1}{e^x}}dy+\mathcal{O}(dx·dy)$$

The differential $dy$ is linearly dependent on $dx$, hence the error $\mathcal{O}(dx·dy)$ is also $\mathcal{O}\left(d{y}^2\right)$. Replacing this expression into the equation $d(lny)=dx$, we get:

$$d(lny)={\displaystyle \frac{1}{e^x}}dy+\mathcal{O}\left(d{y}^2\right)$$

Remembering that $y=e^x$, we replace ${\displaystyle \frac{1}{e^x}}$ with ${\displaystyle \frac{1}{y}}$:

$$d(lny)={\displaystyle \frac{1}{y}}dy+\mathcal{O}\left(d{y}^2\right)$$

Therefore, applying Axiom 8:

$${\displaystyle \frac{d}{dy}}(lny)\equiv \mathrm{Sh}\left({\displaystyle \frac{d(lny)}{dy}}\right)=\mathrm{Sh}\left({\displaystyle \frac{\frac{1}{y}dy+\mathcal{O}\left(d{y}^2\right)}{dy}}\right)=\mathrm{Sh}\left({\displaystyle \frac{1}{y}}+\mathcal{O}\left(dy\right)\right)={\displaystyle \frac{1}{y}}$$

□

The proof of the above theorem can be stated in general terms by proving that the derivative of the inverse function $x=g\left(y\right)$ of a function $y=f\left(x\right)$ is the reciprocal of the derivative $f^{\prime }\left(x\right)$:

$$g^{\prime }\left(y\right)={\displaystyle \frac{1}{f^{\prime }\left(x\right)}}$$

4.8. The Fundamental Theorem of Calculus

Theorem 12 

(FTC). Let $f\left(x\right)$ be a continuous function such that there exists a primitive $F\left(x\right)$ satisfying $dF\left(x\right)=f\left(x\right)dx+\mathcal{O}\left(d{x}^2\right)$. Then ${\int }_a^{b}f\left(x\right)dx=F\left(b\right)-F\left(a\right)$.

Proof. 

Partition the interval $[a,b]$ into $N={\displaystyle \frac{b-a}{dx}}$ steps, where $x_k=a+k·dx$. The constructive integral is defined as ${\int }_a^{b}f\left(x\right)dx\equiv \mathrm{Sh}\left({\sum }_{k=0}^{N-1}f\left(x_k\right)dx\right)$. Substituting $f\left(x_k\right)dx=dF\left(x_k\right)-\mathcal{O}\left(d{x}^2\right)$ from Axiom 6 yields:

$${\displaystyle \sum _{k=0}^{N-1}}f\left(x_k\right)dx={\displaystyle \sum _{k=0}^{N-1}}dF\left(x_k\right)-{\displaystyle \sum _{k=0}^{N-1}}\mathcal{O}\left(d{x}^2\right)$$

(61)

The first summation is a literal telescoping sum: ${\sum }_{k=0}^{N-1}(F\left(x_{k+1}\right)-F\left(x_k\right))=F\left(x_N\right)-F\left(x_0\right)=F\left(b\right)-F\left(a\right)$. The remainder summation scales as $N·\mathcal{O}\left(d{x}^2\right)={\displaystyle \frac{b-a}{dx}}·\mathcal{O}\left(d{x}^2\right)=\mathcal{O}\left(dx\right)$. Applying Sh (Axiom 8) drops the fractional remainder, leaving $F\left(b\right)-F\left(a\right)$. □

5. Multivariable Extension and Jacobian Notation

The architecture of CDC maps symmetrically into multi-dimensional spaces without any modification to the fundamental axioms. Given independent variables x and y, they are assigned their own specific intrinsic infinitesimal seeds: $dx={\left\{d{x}_n\right\}}_{n\in \mathbb{N}}$ and $dy={\left\{d{y}_m\right\}}_{m\in \mathbb{N}}$. The total differential is defined via Axiom 2 as $df(x,y)\equiv f(x+dx,y+dy)-f(x,y)$. Grouping the linear terms yields $df(x,y)=A(x,y)dx+B(x,y)dy+\mathcal{O}(d{x}^2,d{y}^2,dxdy)$, where the partial derivatives are defined simply as the pure linear coefficients: ${\displaystyle \frac{\partial f}{\partial x}}\equiv A(x,y)$ and ${\displaystyle \frac{\partial f}{\partial y}}\equiv B(x,y)$.

5.1. Independent Seeds and the Total Differential

Following Axiom 1, we assign to each independent variable its own intrinsic structural seed:

$$dx=\left\{d{x}_n\right\},dy=\left\{d{y}_m\right\}$$

(62)

For a multivariable function $f(x,y)$, Axiom 2 defines the total differential as the simultaneous increment over these independent microscopic steps:

$$df(x,y)\equiv f(x+dx,y+dy)-f(x,y)$$

(63)

The expression evaluates to a multivariate polynomial structure in $dx$ and $dy$.

5.2. Algebraic Definition of Partial Derivatives (∂)

Grouping the terms of the differential expansion $df(x,y)$ based on the linear components of the generators yields:

$$df(x,y)=A(x,y)dx+B(x,y)dy+\mathcal{O}(d{x}^2,d{y}^2,dxdy)$$

(64)

We define the partial derivatives as the macroscopic coefficients of these pure linear monomials:

$${\displaystyle \frac{\partial f}{\partial x}}\equiv A(x,y),{\displaystyle \frac{\partial f}{\partial y}}\equiv B(x,y)$$

(65)

The curly ∂ notation loses any mystical ambiguity; it simply denotes a projection onto a specific generator within a multivariate algebraic ring. The standard total differential is recovered via the global Shadow operator (Axiom 8):

$$\mathrm{Sh}\left(df\right)={\displaystyle \frac{\partial f}{\partial x}}dx+{\displaystyle \frac{\partial f}{\partial y}}dy$$

(66)

5.3. Multivariable Application Example

Consider $f(x,y)=x^{2}y$. Direct application of the operational identity yields:

$$\begin{array}{cc}\hfill df(x,y)& ={(x+dx)}^2(y+dy)-x^{2}y\hfill \\ \hfill & =(x^2+2xdx+d{x}^2)(y+dy)-x^{2}y\hfill \\ \hfill & =x^{2}y+x^{2}dy+2xydx+2xdxdy+yd{x}^2+d{x}^{2}dy-x^{2}y\hfill \\ \hfill & =\left(2xy\right)dx+\left(x^2\right)dy+\left[2xdxdy+yd{x}^2+d{x}^{2}dy\right]\hfill \end{array}$$

(67)

Substituting the structural sequences shows that the terms inside the brackets belong to higher-order fractional scales ($\mathcal{O}(1/n^2,1/m^2,1/nm)$). The linear coefficients immediately isolate the exact partial derivatives without any limit processes:

$${\displaystyle \frac{\partial f}{\partial x}}=2xy,{\displaystyle \frac{\partial f}{\partial y}}=x^2$$

(68)

The symmetry of mixed partial derivatives (Clairaut’s Theorem) follows trivially from the commutativity of the polynomial generators ($dx·dy=dy·dx$).

6. Differential Equations in ${\mathcal{D}}^{+}$

In the classical paradigm, an ordinary differential equation (ODE) of the form $y^{\prime }=f(x,y)$ is an implicit statement about limits. Within the CDC framework, this structure undergoes an algebraic transmutation. By translating the operational derivative into its fundamental differential components, any classical ODE can be rigorously rewritten as an exact algebraic identity within the non-Archimedean field:

$$dy=f(x,y)dx+O\left(d{x}^{1+a}\right),\mathrm{for}a>0.$$

(69)

Here, the total variation $dy$ is not a fictional linear approximation, but a fully structured constructive sequence in ${\mathcal{D}}^{+}$.

Consequently, solving a differential equation in the CDC does not require tracking abstract limit trajectories in function spaces. Instead, it amounts to finding a sequence solution within the algebraic scale of ${\mathcal{D}}^{+}$ such that the application of the linear Shadow operator perfectly filters out the higher-order remainder:

$$\mathrm{Sh}\left({\displaystyle \frac{dy}{dx}}\right)=\mathrm{Sh}\left(f(x,y)+O\left(d{x}^a\right)\right)=f(x,y).$$

(70)

This approach unifies continuous differential dynamics and discrete computational algorithms (such as finite-difference methods) into a single, cohesive algebraic language, in which numerical truncation errors are structurally internalized as higher-order infinitesimal scales.

6.1. Hyperbolic Conservation Laws

A long-standing limitation of classical differential calculus appears in the study of non-linear hyperbolic conservation laws, traditionally exemplified by Burgers’ equation [30]:

$${\displaystyle \frac{\partial u}{\partial t}}+u{\displaystyle \frac{\partial u}{\partial x}}=0.$$

(71)

This foundational model describes the convection of a fluid amplitude field $u(x,t)$ (x propagation direction, t time), where the local propagation speed depends directly on the amplitude. Consequently, higher values of u travel faster than lower values, causing the profile of a smooth initial amplitude to progressively steepen over time.

This steepening process culminates at a finite time $t^{*}$—known as the breaking time—where $u(x,t)$ develops a vertical gradient. At this localized wavefront, the spatial derivative ${\displaystyle \frac{\partial u}{\partial x}}$ approaches infinity, collapsing the classical definition of pointwise differentiability. To preserve the validity of the conservation law, classical physics is forced to abandon strong differential equations, transitioning instead to weak formulations defined via integral test functions and the distributional Rankine-Hugoniot jump conditions [17,30].

The CDC architecture structurally bypasses this dualism. Consider a shock localization at $x_s$ with a jump $\Delta u=u^{+}-u^{-}$. In ${\mathcal{D}}^{+}$, the spatial variation across the shock along the constructive seed $dx$ is evaluated directly at a given time t as an exact algebraic finite difference:

$${\partial }_{x}u=u(x_s+dx,t)-u(x_s,t)=\Delta u.$$

(72)

The operational partial derivative inside the shock layer is therefore defined as:

$${\displaystyle \frac{\partial u}{\partial x}}=\Delta u·d{x}^{-1}.$$

(73)

Since $dx\in {\mathcal{D}}^{+}$ is a valid vanishing sequence, its algebraic inverse $d{x}^{-1}$ is a well-defined trans-infinite quantity within the non-Archimedean field. Thus, the partial derivative does not cease to exist; it simply transits from the standard real subdomain to the hyper-real scaling layer.

In this singular context, the pointwise valuation of the derivative is formally sustained by extending the Sh operator. When evaluated over a “microscopic” polynomial in $dx$ featuring negative exponents, the extended operator returns the dominant monomial of lowest algebraic degree, which structurally corresponds to the leading transfinite scale.

Evaluating conservation integrals across the shock, the CDC eliminates the need for test functions and weak formulations. The total regularized flux balance is achieved through direct algebraic cancellation of the microscopic scales:

$$\mathrm{Sh}\left({\displaystyle \frac{\partial u}{\partial x}}·dx\right)=\mathrm{Sh}\left(\Delta u\right)=\Delta u$$

(74)

The Rankine-Hugoniot jump conditions emerge naturally as standard algebraic identities derived from the internal microstructure of the shock in ${\mathcal{D}}^{+}$, establishing that the CDC unifies strong and weak solutions under a single, globally computable differential language [8,17].

In this regard, CDC differs from Robinson’s Non-Standard Analysis (NSA). While NSA provides a rigorous logical foundation for infinitesimals via model theory, its reliance on the non-constructive Transfer Principle and non-constructive Ultrafilters (dependent on the Axiom of Choice) prevents it from resolving operational singularities like hyperbolic shocks. Under the Transfer Principle, a function that is non-differentiable in the standard real domain remains strictly non-differentiable in the hyper-real extension.

Conversely, CDC follows a constructive, sequence-based algebraic field ${\mathcal{D}}^{+}$. Differentiability is structurally internalized: the ratio ${\displaystyle \frac{du}{dx}}$ across a shock layer exists explicitly as a well-defined, computable transfinite sequence within the field. By replacing abstract logical equivalence with concrete algebraic scale envelopes ($O\left(d{x}^a\right)$), the CDC provides an environment where singularities are smoothly processed through direct algebraic mechanics, in the spirit of a constructive analysis [4].

6.2. The Dirac $\delta$ without Distributions

In classical mathematical physics, singular objects such as the Dirac delta function $\delta \left(x\right)$ cannot be defined as pointwise functions. Instead, they require the architectural scaffolding of Schwartz distribution theory, where $\delta \left(x\right)$ is defined strictly as a continuous linear functional acting on a space of smooth test functions $\varphi \left(x\right)\in {\mathcal{C}}_c^{\infty }\left(\mathbb{R}\right)$ via the pairing [7,25,26]:

$${\int }_{-\infty }^{+\infty }\delta \left(x\right)\varphi \left(x\right)dx=\varphi \left(0\right).$$

(75)

This topological formulation strips the Delta object of any localized pointwise meaning, rendering non-linear operations—such as the squaring of a distribution $\delta {\left(x\right)}^2$—notoriously ill-defined [26].

Within the Constructive Differential Calculus (CDC) framework, this limitation is bypassed. By leveraging the microscopic texture of the field ${\mathcal{D}}^{+}$, generalized functions are internalized as legitimate, pointwise-computable functions mapping into the non-Archimedean domain.

Let $dx\in {\mathcal{D}}^{+}$ be a fixed constructive differential seed. We define the structural representation of the Dirac delta function, denoted as ${\delta }_{dx}\left(x\right)$, by assigning its base and its transfinite height directly to values in the field ${\mathcal{D}}^{+}$:

$${\delta }_{dx}\left(x\right)=\left\{\begin{array}{cc}d{x}^{-1}\hfill & \mathrm{if}0\le x<dx,\hfill \\ 0\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

(76)

Unlike classical analysis, where taking the limit $dx\to 0$ collapses the function, ${\delta }_{dx}\left(x\right)$ remains a well-defined algebraic mapping for any non-zero $dx\in {\mathcal{D}}^{+}$.

To demonstrate its operational validity, consider a continuous function $\varphi \left(x\right)$. Through the fundamental structural decomposition of the CDC, $\varphi \left(x\right)$ can be expanded in the microscopic neighborhood of the origin along the scale envelopes of the seed:

$$\varphi \left(x\right)=\varphi \left(0\right)+{\varphi }^{\prime }\left(0\right)x+\mathcal{O}\left(x^2\right).$$

(77)

The action of the constructive delta function under a microscopic integral partition is computed through direct algebraic pairing within ${\mathcal{D}}^{+}$:

$$I={\int }_{-\infty }^{+\infty }{\delta }_{dx}\left(x\right)\varphi \left(x\right)dx={\int }_0^{dx}d{x}^{-1}\left[\varphi \left(0\right)+{\varphi }^{\prime }\left(0\right)x+O\left(x^2\right)\right]dx.$$

(78)

By invoking the linearity of the integration operator over the algebraic grid of the field, we distribute the transfinite scaling factor $d{x}^{-1}$:

$$I=d{x}^{-1}{\int }_0^{dx}\varphi \left(0\right)dx+d{x}^{-1}{\int }_0^{dx}{\varphi }^{\prime }\left(0\right)xdx+d{x}^{-1}{\int }_0^{dx}O\left(x^2\right)dx.$$

(79)

Evaluating these elementary polynomial terms yields an exact identity inside ${\mathcal{D}}^{+}$ prior to any macroscopic projection:

$$I=d{x}^{-1}\left[\varphi \left(0\right)dx\right]+d{x}^{-1}\left[{\varphi }^{\prime }\left(0\right){\displaystyle \frac{d{x}^2}{2}}\right]+d{x}^{-1}\left[\mathcal{O}\left(d{x}^3\right)\right]=\varphi \left(0\right)+{\displaystyle \frac{1}{2}}{\varphi }^{\prime }\left(0\right)dx+\mathcal{O}\left(d{x}^2\right).$$

(80)

The transfinite height $d{x}^{-1}$ and the infinitesimal base $dx$ undergo exact algebraic cancellation within the first structural shell. Finally, applying the linear Shadow operator (Sh) isolates the macroscopically observable physical residue:

$$\mathrm{Sh}\left(I\right)=\mathrm{Sh}\left(\varphi \left(0\right)+{\displaystyle \frac{1}{2}}{\varphi }^{\prime }\left(0\right)dx+O\left(d{x}^2\right)\right)=\varphi \left(0\right).$$

(81)

This confirms that the classical distributional behavior emerges naturally as a projection of a deterministic, term-by-term algebraic mechanics.

6.3. The Electron Self-Energy Divergence and CDC Renormalization

A classic manifestation of singular breakdown in field theories occurs when evaluating the electrostatic self-energy of a point-like particle. In classical electrodynamics, modeling an electron as a sphere of total charge e and radius R yields a field-energy integral of:

$$E_{\mathrm{field}}\left(R\right)={\int }_R^{\infty }{\displaystyle \frac{1}{2}}{ϵ}_0{\left|\mathbf{E}\right|}^{2}dV={\displaystyle \frac{e^2}{8\pi {ϵ}_{0}R}}.$$

(82)

Taking the classical point-like limit ($Li{m}_{R\to 0}$) forces the energy to diverge to infinity, implying an infinite effective mass ($E=m{c}^2$) for a physical particle known to possess a finite, minuscule observable mass. To circumvent this, standard renormalization theory [9,29] postulates an unobservable, infinitely negative "bare mass" ($m_0$) to execute a formal subtraction of the type $\infty -\infty$, an operation structurally invalid within standard real analysis [7].

Within the CDC framework, this problem is internalized directly by setting the physical radius of the point charge equal to the linear microscopic seed, $R=dx\in {\mathcal{D}}^{+}$. Let us define the macroscopic physical constant $C={\displaystyle \frac{e^2}{8\pi {ϵ}_0}}$. The field energy is mapped onto a well-defined, structured trans-infinite element of the field:

$$E_{\mathrm{field}}\left(dx\right)=C·d{x}^{-1}.$$

(83)

Simultaneously, the bare energy of the particle $E_{\mathrm{bare}}$ is not a static real number but an active algebraic object inside ${\mathcal{D}}^{+}$, whose functional dependence on the microscopic cutoff scale $dx$ is explicitly structured as:

$$E_{\mathrm{bare}}\left(dx\right)=E_{\mathrm{std}}-C·d{x}^{-1}+O\left(dx\right),$$

(84)

where $E_{\mathrm{std}}$ represents the finite, standard physical energy observed macroscopically.

The total energy of the physical system, $E_{\mathrm{tot}}$, is computed via direct algebraic addition within the non-Archimedean field ${\mathcal{D}}^{+}$ prior to any limit processing or external projection:

$$E_{\mathrm{tot}}\left(dx\right)=E_{\mathrm{bare}}\left(dx\right)+E_{\mathrm{field}}\left(dx\right)=\left[E_{\mathrm{std}}-C·d{x}^{-1}+O\left(dx\right)\right]+\left[C·d{x}^{-1}\right].$$

(85)

By invoking the field properties of ${\mathcal{D}}^{+}$, specifically associativity and commutativity over sequence-generated components, the transfinite scaling terms of identical order are directly paired:

$$E_{\mathrm{tot}}\left(dx\right)=E_{\mathrm{std}}+\left(C·d{x}^{-1}-C·d{x}^{-1}\right)+O\left(dx\right).$$

(86)

Because $d{x}^{-1}$ is a legitimate algebraic element executing an exact term-by-term inversion on the underlying sequence, the subtraction yields a precise cancellation within the field:

$$C·d{x}^{-1}-C·d{x}^{-1}=0\Rightarrow E_{\mathrm{tot}}\left(dx\right)=E_{\mathrm{std}}+O\left(dx\right).$$

(87)

Finally, the application of the Shadow operator (Sh) filters out the remaining infinitesimal fluctuations, isolating the standard physical residue:

$$\mathrm{Sh}\left(E_{\mathrm{tot}}\left(dx\right)\right)=\mathrm{Sh}\left(E_{\mathrm{std}}\right)+\mathrm{Sh}\left(O\left(dx\right)\right)=E_{\mathrm{std}}$$

(88)

This mathematical analysis demonstrates that renormalization is not a heuristic subtraction of logical infinities, but a native, algebraic scale alignment. Robinson’s NSA cannot replicate this exact pairing because its model-theoretic iperreals lack a constructive sequence tracker ($O\left(dx\right)$ modules); a generic non-standard infinity $\Omega$ cannot be deterministically locked to a specific geometric spatial scaling, causing the subtraction $\Omega -\Omega$ to produce an unpredictable, non-unique standard part (see [1] for related non constructive solutions via non-standard analysis).

6.4. Limitations of Robinson’s NSA

It is mathematically instructive to articulate why the precise constructions of the subsections above cannot be replicated within Robinson’s Non-Standard Analysis (NSA), highlighting the structural departure of the CDC from model-theoretic hyperreals [1].

First, Robinson’s framework relies on the non-constructive Transfer Principle. If a function is bounded or fails to possess a pointwise divergence in the standard real domain $\mathbb{R}$, its non-standard extension strictly preserves that qualitative behavioral classification. NSA cannot dynamically assign a polynomial micro-structure to a singularity because it lacks an inherent algebraic calculus of scale envelopes—such as a CDC $O\left(d{x}^a\right)$ module. In NSA, an infinitesimal is generic; it does not carry a deterministic sequence index capable of tracking higher-order interactions during integration.

Second, the lack of a constructive generator introduces a profound foundational barrier regarding the algebraic consistency of non-linear operations on singularities. In Robinson’s NSA, an infinitesimal $\eta =\left[\left\{{\eta }_n\right\}\right]$ is an equivalence class of sequences defined modulo a non-constructive Free Ultrafilter$\mathcal{U}$. While the algebraic field axioms of $*\mathbb{R}$ legally permit operations such as squaring the inverse (η⁻²), a severe consistency issue arises because highly non-linear transformations do not commute with the ultrafilter-driven identification process.

The ultrafilter acts as a non-constructive, measure-like selection mechanism that preserves first-order linear structures. However, when evaluating non-linear operations on singular objects (such as the squared Dirac delta), the structural representation of the transfinite residue becomes path-dependent, relying strictly on the specific choice of $\mathcal{U}$.

Consequently, two distinct but equally valid free ultrafilters, ${\mathcal{U}}_1$ and ${\mathcal{U}}_2$, partition the underlying sequence space differently, yielding mutually incompatible and non-transferable representations of the squared singularity upon projection via the standard part operator (st). The CDC architecture, conversely, bypasses this bottleneck entirely: because the seed $dx\in {\mathcal{D}}^{+}$ is an explicit, constructive sequence, its algebraic powers ($d{x}^2,d{x}^{-1},d{x}^{-2}$) are rigidly ordered, deterministic scales within the non-Archimedean field, ensuring that non-linear operations on singularities are globally unique, trackable, and independent of any non-constructive choice.

7. Conclusions

The Constructive Differential Calculus (CDC) systematizes Leibniz’s intuitive calculus within eight simple algebraic postulates. We proved the core rules of derivatives, integrals, and power expansions through polynomial and analytic operators over the field of real constant sequences extended by the definitely nonzero infinitesimals and their inverses, generated by basic differentials. CDC offers a rigorous alternative to logical and algebraic approaches based on non-standard analysis and ultrafilters, transforming calculus into a transparent extension of elementary algebra.

To elevate our algebraic structure from a ring to a field, the introduction of infinite quantities is not an arbitrary choice, but a logical necessity driven by the field axiom of multiplicative inversion. If an x-infinitesimal $dx$ is definitely nonzero, its multiplicative inverse must exist within the field (${\mathbb{N}}^{+}$, the positive naturals):

$$dx={\left\{{\displaystyle \frac{1}{n}}\right\}}_{n\in {\mathbb{N}}^{+}}={\Omega }^{-1}$$

that is:

$$\Omega =d{x}^{-1}={\displaystyle \frac{1}{dx}}={\left\{n\right\}}_{n\in \mathbb{N}}$$

In this framework, the product annihilation property is satisfied in full because we do not admit independent, disconnected zeros in the seed sequences.

Structurally, the non-Archimedean field ${\mathcal{D}}^{+}$ generated by the CDC framework can be viewed as a constructive, partially ordered subfield of Robinson’s hyperreals ($*\mathbb{R}$). While Robinson restores the totality of ordering over arbitrary vanishing sequences by invoking non-constructive logical expedients—namely, non-principal ultrafilters dependent on the Axiom of Choice—the CDC deliberately constructs differential starting from $dx$ seeds by using algebraic operations and iterated differential operators, with related Taylor-McLaurin expansions.

The "ghosts" of Berkeley are thus framed within a balanced algebraic ecosystem, in which infinity and the infinitesimal are simply two perspectives on the same recursive sequence.

The algebraic robustness of CDC is deeply rooted in the analytical tameness of its admissible functions. By restricting our domain to the class of logarithmic-exponential functions originally introduced by Hardy [10,11], we ensure that the germs of our variations form a differential field—traditionally designated as a Hardy field following Bourbaki [5]. The non-oscillatory nature of these functions, extensively formalized by Rosenlicht [24], guarantees that every differential $df\left(x\right)$ collapses to its local monomial dominance [28], thereby eliminating any pathological zero-crossings in the neighborhood of the core infinitesimal $dx$.

As established in our formulation of Theorem 2, the total ordering of asymptotic scales within a Hardy field [5,24] ensures that every variation collapses to a dominant monomial of the form $c·d{x}^{\alpha }$. This rigid structural tameness eliminates any pathological, oscillatory zero-crossings, thereby validating the Nonzero Definiteness of the core differential $df\left(x\right)$.

A spontaneous epistemological question arises: Given that Hardy possessed all the necessary asymptotic machinery, why did he not conceive the CDC himself? Why did he limit his Infinitärcalcül to a descriptive calculus of growth orders rather than turning it into an operational calculus of actual, standalone infinitesimals?

The answer is historical and foundational, and can be summarized in three distinct epistemological barriers that only modern mathematics has been able to bridge:

1.

The Foundational Terror of the 19th Century: Hardy operated in an era completely dominated by the Weierstrassian triumphalism of the $ϵ$-$\delta$ formalism. Infinitesimals had been forcefully banished from analysis as logical chimeras. In 1910, any attempt to define $dx$ as an actual, non-zero object outside of a limit fraction would have been rejected as a regression to pre-Cauchy vagueness. Hardy’s goal was to make du Bois-Reymond’s ideas respectable to the mathematical establishment of his time, which meant he had to frame his calculus strictly through standard limits of functions tending to infinity.

2.

The Absence of Model Theory: Abraham Robinson’s Nonstandard Analysis [23]—which proved that non-Archimedean fields could be as rigorous as $\mathbb{R}$—would not appear until 1960. Furthermore, the concept of a Hardy field as a differential algebra was only formalized by Bourbaki in 1961 [5]. Hardy viewed his log-exponential scales as a hierarchy of functions, not as elements of a valued, non-Archimedean ordered differential field.

3.

Descriptive vs. Operational Intent: Hardy’s primary mathematical interest in Orders of Infinity was classification, not derivation. He sought a universal scale to measure how fast a function diverges or converges—a thermometer for asymptotic growth. CDC, conversely, is an operational shift: it takes that asymptotic scale and forces it down into the differentials of a point, treating the generator $dx$ not as an elusive variable tending to zero, but as a fixed, foundational basic component.

CDC can be seen as pairing the algebraic robustness of Hardy fields with a modern, non-Archimedean handling of infinitesimals by providing a rigorous framework where the ghosts of differentials are finally materialized into well-behaved, non-zero algebraic entities.

For nearly two centuries, the introduction of calculus has been burdened by the classical $ϵ$-$\delta$ formulation. While undeniably rigorous, this approach introduces a severe cognitive rupture: it forces students to abandon their natural geometric intuition of dynamic variations in favor of static, counterintuitive alternating quantifiers ($\forall ϵ>0,\exists \delta >0$). This formalism often acts as a pedagogical barrier, obscuring the physical and geometric essence of derivatives and integrals under a cloud of nested inequalities.

The CDC framework restores the historical, intuitive workflow of the pioneers of calculus without sacrificing mathematical rigor. By replacing the non-constructive limit process with an algebraic operational core—shielded by the Algebraic Closure of Theorem 2 and the behavior of the Shadow operator (Sh) at a fixed point—differentiation is returned to its rightful status: a clean, algorithmic extraction of the dominant monomial scale. Students can once again manipulate differentials as active, non-zero microscopic entities, executing Taylor-based algebraic expansions (such as Newton’s binomial series) with absolute logical safety.

In this sense, an apt and historically faithful alternative title for the present formulation could easily be:

“Leibniz ab omni naevo vindicatus”

Borrowing the spirit of Saccheri’s famous foundational attempt, this work delivers a systematic vindication of Gottfried Wilhelm Leibniz’s original intuition. By clearing his characteristically dynamic differentials from the historical “stain” (naevus) of logical inconsistency, CDC proves that the infinitesimals of the Acta Eruditorum merely require a constructive, sequence-based screening mechanism giving the explicit foundations of the original differential calculus.