Hierarchical Derivation Rules

Ahmed Gossa
Independent Researcher — Hierarchical Calculus

This page summarizes the operational rules of hierarchical differentiation. We begin with the relative derivative (rank 1) then generalize to rank \(n\).
Notation rule: every derivative is written with explicit rank and degree: \(\;D_r^n\). In this page we use degree 1 only: \(\;D_{0}^{1}, D_{1}^{1}, D_{n}^{1}\).

Concept DOI (main reference): 10.5281/zenodo.17917302

Version DOI (v1.0.0): 10.5281/zenodo.17917303 — website release v1.0.0

Domain note (degree 1): For rank 1 we typically require \(x>0\) and \(f(x)>0\). For rank 2 we typically require \(x>1\) and \(f(x)>1\).

0) Quick intuition (for all readers)

Classical calculus measures absolute change using \(D_0^1 f=\frac{df}{dx}\). Hierarchical calculus adds higher ranks that measure change on transformed scales:

Rank 1 measures relative change (log-log scale): \[ D_1^1 f = \frac{d\ln f}{d\ln x}. \] If this quantity is constant, the function follows a power law. This is why rank 1 is important in physics, engineering, and scaling theory.

One sentence summary

Rank increases when absolute variation becomes less informative than scale-invariant variation.

1) Quick definition: the relative derivative (rank 1, degree 1)

Assume \(x>0\) and \(f(x)>0\). The relative derivative measures change on a relative (ratio) scale:

\[ \boxed{ D_{1}^{1} f(x) =\frac{d\ln f(x)}{d\ln x} } \]

1.1) Link with the classical derivative (rank 0, degree 1)

The classical derivative is \(D_0^{1}f=f'(x)\). The relative derivative can be written as:

\[ \boxed{ D_{1}^{1} f(x) = \frac{x}{f(x)}\,\frac{df}{dx} = \frac{x\,D_0^{1}f(x)}{f(x)} } \]

2) Core rules for \(D_{1}^{1}\)

2.1) Multiplication by a constant

\[ \boxed{ D_{1}^{1}(C\,f)=D_{1}^{1} f \qquad (C>0\ \text{constant}) } \]

2.2) Product rule

\[ \boxed{ D_{1}^{1}(fg)=D_{1}^{1} f + D_{1}^{1} g } \]

2.3) Quotient rule

\[ \boxed{ D_{1}^{1}\!\left(\frac{f}{g}\right)=D_{1}^{1} f - D_{1}^{1} g } \]

2.4) Constant exponent

\[ \boxed{ D_{1}^{1}\!\left(f^a\right)=a\,D_{1}^{1} f \qquad (a\ \text{constant}) } \]

2.5) Power function

\[ \boxed{ D_{1}^{1}(x^\alpha)=\alpha } \]

2.6) Chain rule — relative form

Let \(u=u(x)>0\) and \(f=f(u)>0\). Then:

\[ \boxed{ D_{1}^{1}\big(f(u(x))\big) =\left(\frac{d\ln f}{d\ln u}\right)\Bigg|_{u=u(x)} \cdot \frac{d\ln u}{d\ln x} } \]

3) Examples (worked)

3.1) Example: \(f(x)=x^a\)

\[ D_1^1(x^a)=a. \]

Meaning: power laws appear as constants in rank 1. This is why rank 1 is the natural scale for self-similar processes.

3.2) Example: \(f(x)=e^x\)

\[ D_1^1(e^x)=x. \]

3.3) Example: \(f(x)=\ln x\)

\[ D_1^1(\ln x)=\frac{1}{\ln x}. \]

4) Self-check (mini exercises)

Try to compute \(D_1^1 f\) in each case:

\(f(x)=x^5\)
\(f(x)=x^{-2}\)
\(f(x)=\sqrt{\ln x}\)
\(f(x)=\dfrac{x^2}{\ln x}\)

Hint:

Use: product rule + quotient rule + constant exponent rule + chain rule.

5) Generalization to rank \(n\): \(D_n^1\)

\[ \boxed{ D_{n}^{1} f(x) =\frac{d\,\ln^{(n)}(f(x))}{d\,\ln^{(n)}(x)} } \]

Key idea: The same structural rules remain valid for all ranks as long as the logarithmic chains are defined.

\[ \boxed{ D_{n}^{1}(fg)=D_{n}^{1} f + D_{n}^{1} g,\qquad D_{n}^{1}\!\left(\frac{f}{g}\right)=D_{n}^{1} f - D_{n}^{1} g,\qquad D_{n}^{1}(f^a)=a\,D_{n}^{1} f } \]

6) FAQ (short)

6.1) Why do we use logs?

Logs convert multiplicative changes into additive ones. This makes scaling rules simple and stable.

6.2) When is rank 1 better than rank 0?

When you care about percentage / ratio growth rather than absolute growth.

6.3) What is the meaning of higher ranks?

Higher ranks measure changes of the growth mechanism itself (nested scale behavior).

7) Related topics

Relative derivative, logarithmic derivative, iterated logarithms, scaling laws, self-similarity, power-law behavior, hierarchical derivatives, hierarchical equations.

8) Citation

\[ \texttt{GOSSA AHMED. Hierarchical Calculus — Derivation Rules (D\_1 then D\_n, explicit rank \& degree). Official Website. Zenodo. https://doi.org/10.5281/zenodo.17917302 (2025).} \]