Relationships between hierarchical derivatives
Ahmed Gossa
Independent Researcher — Hierarchical Calculus
Hierarchical derivatives are not independent operators; they form linked levels of a single derivative pyramid.
Notation rule: every derivative is written with both rank and degree: \(\;D_n^k\).
This page mainly concerns degree 1, so we write:
\(\;D_0^{1}, D_1^{1}, D_2^{1}, \dots, D_n^{1}\).
1) General definition (degree 1)
For \(y=y(x)\), the rank-\(n\), degree-1 derivative can be written as:
\[
\boxed{
D_{n}^{1} y
= \frac{dy}{dx}\;\frac{x\,\phi_n(x)}{y\,\phi_n(y)}
}
\]
Example scaling functions:
\[
\phi_1(t)=1,\quad \phi_2(t)=\ln t,\quad \phi_3(t)=\ln(\ln t),\;\dots
\]
2) First ranks (degree 1)
\[
D_{0}^{1}y=\frac{dy}{dx},
\qquad
D_{1}^{1}y=\frac{dy\cdot x}{dx\cdot y},
\qquad
D_{2}^{1}y=\frac{dy\cdot x\ln x}{dx\cdot y\ln y}.
\]
3) Master link to \(D_{0}^{1}\)
\[
\boxed{
D_{n}^{1}y = D_{0}^{1}y \;\frac{x\,\phi_n(x)}{y\,\phi_n(y)}
}
\]
4) Consecutive ranks: \(D_{n}^{1} \leftrightarrow D_{n+1}^{1}\)
\[
\boxed{
D_{n+1}^{1} y
= D_{n}^{1} y \;\frac{\phi_{n+1}(x)}{\phi_n(x)} \;\frac{\phi_n(y)}{\phi_{n+1}(y)}
}
\]
5) Direct relations (degree 1)
-
From \(D_{0}^{1}\) to \(D_{1}^{1}\):
\[
\boxed{
D_{1}^{1}y = D_{0}^{1}y\;\frac{x}{y}
}
\]
-
From \(D_{1}^{1}\) to \(D_{2}^{1}\):
\[
\boxed{
D_{2}^{1}y = D_{1}^{1}y\;\frac{\ln x}{\ln y}
}
\]
-
From \(D_{2}^{1}\) to \(D_{3}^{1}\):
\[
\boxed{
D_{3}^{1}y = D_{2}^{1}y\;\frac{\ln(\ln x)}{\ln(\ln y)}
}
\]
6) Compact rank ladder (degree 1)
\[
\boxed{
D_{n}^{1}y = D_{1}^{1}y \prod_{k=2}^{n} \frac{\phi_k(x)}{\phi_k(y)}
}
\]
\[
\boxed{
D_{n}^{1}y = D_{0}^{1}y \;\frac{x}{y} \prod_{k=2}^{n} \frac{\phi_k(x)}{\phi_k(y)}
}
\]
Note:
Higher degrees (e.g. \(D_1^{2}\), \(D_1^{3}\), …) are obtained by successive application within the same rank,
as defined on the “Rank & Degree” page.
