Every derivative has a rank and a degree

Ahmed Gossa
Independent Researcher — Hierarchical Calculus

In Hierarchical Calculus, a derivative is never written without specifying its rank (type / transformation layer) and its degree (how many successive applications). The compact notation is:

\[ \boxed{D_{r}^{n}} \]

where \(r \in \{0,1,2,\dots\}\) is the rank (0: differential, 1: relative, 2: logarithmic, …) and \(n \in \{0,1,2,\dots\}\) is the degree.

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

Related pages: Derivative Relationships · Rules

Single-source-of-truth rule

Throughout this site, the rank-\(r\) derivative is defined by applying the same iterated-log transform to both sides (to \(x\) and to \(f(x)\)). This prevents contradictions between pages.

Quick Table of Contents

1) Core definitions (rank, degree, iterated logs) 2) The adopted definition of \(D_r^1\) (degree 1) 3) Degree as successive application 4) Domain conditions (what must be positive) 5) Relations: connecting ranks and computable operators \(\delta_r\) 6) Worked examples (D0 and D1) 7) The rank–degree grid (organizational view) 8) Why this page matters for Series expansions

1) Core definitions

1.1 Iterated logarithms (the hierarchy variables)

Define the iterated logarithm \(\ln^{(r)}\) recursively:

\[ \ln^{(0)}(x)=x,\qquad \ln^{(1)}(x)=\ln(x),\qquad \ln^{(r+1)}(x)=\ln\!\big(\ln^{(r)}(x)\big). \]

We may use the shorthand hierarchical variable: \[ u_r(x) := \ln^{(r)}(x). \] Similarly on the function side: \[ v_r(f(x)) := \ln^{(r)}(f(x)). \]

1.2 What is “rank”?

Rank \(r\) selects the transformation layer. Intuitively, rank changes the scale on which you measure variation:

Rank 0 (D0): variation with respect to \(x\) itself (classical differential calculus).
Rank 1 (D1): variation with respect to \(\ln(x)\) (relative / multiplicative scale).
Rank 2: variation with respect to \(\ln(\ln(x))\) (nested log scale).

1.3 What is “degree”?

Degree \(n\) counts successive applications within the same rank. Degree does not change the rank. You first choose the rank, then you raise the degree by repetition.

2) The adopted definition of \(D_r^1\) (degree 1)

2.1 Unified definition (this is the only definition used on the site)

The rank-\(r\), degree-1 hierarchical derivative is defined by:

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

Consistency note: We do not use mixed forms such as \(\frac{d(\ln f)}{d\ln^{(r)}x}\). The adopted definition applies \(\ln^{(r)}\) to both sides.

2.2 Special cases (D0 and D1)

\[ D_{0}^{1}f=\frac{df}{dx}, \qquad D_{1}^{1}f=\frac{d\ln f}{d\ln x}, \qquad D_{2}^{1}f=\frac{d\ln(\ln f)}{d\ln(\ln x)}. \]

This makes classical calculus appear explicitly as the rank-zero case.

3) Degree as successive application

3.1 Operator definition of degree

Degree \(n\) means applying the same rank derivative \(n\) times:

\[ \boxed{ D_{r}^{n}(\cdot)=\underbrace{D_{r}^{1}\circ D_{r}^{1}\circ\cdots\circ D_{r}^{1}}_{n\ \text{times}}(\cdot) } \]

In particular, \(D_{r}^{0}\) is the identity operator: \[ D_{r}^{0}f=f. \]

3.2 Two useful degree sequences (don’t confuse them)

In computations (especially for series expansions), it is often useful to distinguish:

(A) Degrees applied to \(f\) itself: \(D_r^n f = D_r^1(D_r^{n-1}f)\).
(B) Degrees applied to the transformed quantity: \(A_n^{(r)} := \big(D_r^n\,\ln^{(r)} f\big)\). This is the practical sequence that naturally appears in multiplicative/exponential series.

4) Domain conditions (what must be positive)

4.1 Why domains matter

Because \(\ln^{(r)}\) appears explicitly, you must respect positivity constraints. Typical examples:

\[ r=1:\ x>0,\ f(x)>0. \qquad r=2:\ x>1,\ f(x)>1 \ \ (\text{often, to make }\ln(\ln(\cdot))\text{ defined}). \]

Domain note: The exact domain depends on the rank and the function. The safe rule is: the entire chain entering \(\ln^{(r)}\) must remain in its valid domain.

5) Relations: connecting ranks and computable operators \(\delta_r\)

5.1 Linking \(D_0\) and \(D_1\)

The rank-1 relative derivative connects directly to the standard derivative:

\[ \boxed{ D_{1}^{1}f(x)=\frac{d\ln f}{d\ln x} =\frac{x}{f(x)}\,\frac{df(x)}{dx} =\frac{x}{f(x)}\,D_{0}^{1}f(x) } \]

Hence:

\[ \boxed{ D_{0}^{1}f(x)=\frac{f(x)}{x}\,D_{1}^{1}f(x) } \]

5.2 The computable “rank operator” \(\delta_1\)

Define the scale operator:

\[ \boxed{ \delta_1 := \frac{d}{d\ln x}=x\frac{d}{dx} } \]

Then the relative derivative can be written compactly as:

\[ \boxed{ D_{1}^{1}f=\delta_1(\ln f) } \]

5.3 The rank-2 operator \(\delta_2\)

For rank 2:

\[ \boxed{ \delta_2 := \frac{d}{d\ln(\ln x)} = x\ln(x)\frac{d}{dx} } \]

Then:

\[ \boxed{ D_{2}^{1}f=\delta_2(\ln(\ln f)) } \]

5.4 General pattern for \(\delta_r\)

The general idea is: \(\delta_r\) differentiates with respect to \(\ln^{(r)}(x)\). For practical ranks (0,1,2) we already have explicit forms; higher ranks follow the same principle.

6) Worked examples (D0 and D1)

Example A: \(f(x)=x^a\) (with \(x>0\))

\[ D_0^1(x^a)=a x^{a-1}, \qquad D_1^1(x^a)=\frac{d\ln(x^a)}{d\ln(x)}=a. \]

Rank changes the meaning: \(D_0\) measures classical differential change, while \(D_1\) measures relative (log-scale) change.

Example B: \(f(x)=e^x\) (with \(x>0\) for rank 1)

\[ D_0^1(e^x)=e^x, \qquad D_1^1(e^x)=\frac{d\ln(e^x)}{d\ln(x)}=\frac{d(x)}{d(\ln x)}=x. \]

Example C: Constant \(f(x)=c>0\)

\[ D_0^1(c)=0, \qquad D_1^1(c)=\frac{d\ln(c)}{d\ln(x)}=0. \]

7) The rank–degree grid (organizational view)

A compact symbolic table

\[ \begin{array}{c|ccc} \text{Rank } r & \text{Degree 1} & \text{Degree 2} & \text{Degree 3}\\ \hline 0 & D_0^{1}f & D_0^{2}f & D_0^{3}f\\ 1 & D_1^{1}f & D_1^{2}f & D_1^{3}f\\ 2 & D_2^{1}f & D_2^{2}f & D_2^{3}f\\ \vdots & \vdots & \vdots & \vdots \end{array} \]

This grid enforces precision: first fix the rank, then raise the degree by repetition.

8) Why this page matters for Series expansions

8.1 Practical degree raising (the coefficients used in Series pages)

Within a fixed rank, raising the degree becomes computable through \(\delta_r\). Define:

\[ \delta_1:=\frac{d}{d\ln x}=x\frac{d}{dx}, \qquad \delta_2:=\frac{d}{d\ln(\ln x)}=x\ln(x)\frac{d}{dx}. \]

For rank \(1\), a practical coefficient sequence is: \[ A_n^{(1)}(x) := \big(D_1^n\,\ln f\big)(x). \] Then:

\[ \boxed{ A_{1}^{(1)}(x)=\delta_1(\ln f)=\frac{x}{f(x)}\,\frac{df}{dx} } \]

\[ \boxed{ A_{n+1}^{(1)}(x)=\delta_1\!\left(A_{n}^{(1)}(x)\right) =x\frac{d}{dx}A_{n}^{(1)}(x) } \]

Early degrees (rank \(1\)):

\[ \boxed{ \big(D_{1}^{2}\ln f\big)(x) =x\frac{d}{dx}\!\left(\frac{x f'(x)}{f(x)}\right) } \]

\[ \boxed{ \big(D_{1}^{3}\ln f\big)(x) =x\frac{d}{dx}\!\left[x\frac{d}{dx}\!\left(\frac{x f'(x)}{f(x)}\right)\right] } \]

Similarly for rank \(2\), define \(A_n^{(2)}(x):=\big(D_2^n\,\ln^{(2)}f\big)(x)\) with \(\ln^{(2)}f=\ln(\ln f)\). Then:

\[ \boxed{ A_{1}^{(2)}(x)=\delta_2\!\big(\ln(\ln f)\big) } \]

\[ \boxed{ A_{n+1}^{(2)}(x)=\delta_2\!\left(A_{n}^{(2)}(x)\right) =x\ln(x)\frac{d}{dx}A_{n}^{(2)}(x) } \]

Why this matters

These relations give a direct computational recipe for raising the degree inside the same rank (especially \(D_1\) and \(D_2\)). This is exactly the kind of sequence used for multiplicative/exponential series on the Series pages.

Practical remark: This page is the “glue” that prevents notation drift: it defines rank/degree precisely and provides the computational operators that make the series formulas explicit.