Another approach to get derivative of odd-power

Petro Kolosov Software Developer, DevOps Engineer kolosovp94@gmail.com https://kolosovpetro.github.io

(Date: August 14, 2025)

Abstract.

In this manuscript, we provide and discuss another approach to get derivative of odd-power such that is based on an identity in partial derivatives in terms of polynomial function $f_{y}$ defined as

f_{y}(x,z)=\sum_{k=1}^{z}\sum_{r=0}^{y}\mathbf{A}_{y,r}k^{r}(x-k)^{r}

where $x,z\in\mathbb{R}$ , $y$ is fixed constant $y\in\mathbb{N}$ and $\mathbf{A}_{y,r}$ are real coefficients.

Key words and phrases:

Derivative, Partial derivatives, Partial differential equations, Polynomials

2010 Mathematics Subject Classification:

32W50, 11C08

1. Introduction and Main Results

This manuscript provides another approach to get derivative of odd-power, that is an approach based on partial derivatives of the polynomial function $f_{y}(x,z)$ defined as

f_{y}(x,z)=\sum_{k=1}^{z}\sum_{r=0}^{y}\mathbf{A}_{y,r}k^{r}(x-k)^{r}

where $x,z\in\mathbb{R}$ , $y$ is fixed constant $y\in\mathbb{N}$ and $\mathbf{A}_{y,r}$ are real coefficients. The essence of the approach we discuss is build on an identity in terms of sum of partial derivatives of the polynomial function $f_{y}$ . The function $f_{y}$ is defined by the main results of the manuscript [1] that explains an odd-power in a form as follows

n^{2m+1}=\sum_{k=1}^{n}\sum_{r=0}^{m}{\mathbf{A}}_{m,r}k^{r}(n-k)^{r}

(1)

where $m$ is fixed constant $m\in\mathbb{N}$ , $n\in\mathbb{N}$ and ${\mathbf{A}}_{m,r}$ are real coefficients defined recursively, see [2]. We define the function $f_{y}$ such that based on the identity (1) with the only difference that values of $n,m$ in the right part of (1) appear to be parameters of the function $f_{y}$ . In contrast to the equation (1), upper bound $n$ of the sum $\sum_{k=1}^{n}$ turned into fixed function’s parameter $y$ as well. Let the function $f_{y}$ be defined as follows

Definition 1.1.

(Polynomial function $f_{y}$ .)

f_{y}(x,z)=\sum_{k=1}^{z}\sum_{r=0}^{y}{\mathbf{A}}_{y,r}k^{r}(x-k)^{r}

(2)

where $x,z\in\mathbb{R}$ and $y$ is constant $y\in\mathbb{N}$ . Note that for every $x\in\mathbb{R}$ and constant $y\in\mathbb{N}$ the polynomial identity satisfies

f_{y}(x,x)=x^{2y+1}

At first glance, the definition (2) might look complex, so in order to clarify the function $f_{y}$ and polynomials it produces, let there be a few examples. Substituting the values of $y=1,2,3$ to the function $f_{y}$ we get the following polynomials in $x,z$

	$\displaystyle f_{1}(x,z)$	$\displaystyle=3xz-3z^{2}+3xz^{2}-2z^{3}$
	$\displaystyle f_{2}(x,z)$	$\displaystyle=5x^{2}z-15xz^{2}+15x^{2}z^{2}+10z^{3}-30xz^{3}+10x^{2}z^{3}+15z^{4}-15xz^{4}+6z^{5}$
	$\displaystyle f_{3}(x,z)$	$\displaystyle=-7xz+14x^{2}z+7z^{2}-42xz^{2}+35x^{3}z^{2}+28z^{3}-140x^{2}z^{3}+70x^{3}z^{3}+175xz^{4}$
		$\displaystyle-210x^{2}z^{4}+35x^{3}z^{4}-70z^{5}+210xz^{5}-84x^{2}z^{5}-70z^{6}+70xz^{6}-20z^{7}$

These polynomials are obtained by rearranging the sums in the definition (2) as

f_{y}(x,z)=\sum_{r=0}^{y}{\mathbf{A}}_{y,r}\left[\sum_{k=1}^{z}k^{r}(x-k)^{r}\right]

So that part $\sum_{k=1}^{z}k^{r}(x-k)^{r}$ is polynomial in $x,z$ calculated using Faulhaber’s formula [3]. According to the main topic of the current manuscript, it provides another approach to get derivative of odd-power. Therefore, we define odd-power function we work in the context of. The odd-power function $g_{y}$ is a function defined as follows

Definition 1.2.

(Odd-power function $g_{y}$ .)

g_{y}(x)=x^{2y+1}

where $x\in\mathbb{R}$ and $y$ is constant $y\in\mathbb{N}$ . The Interesting part is that odd-power function $g_{y}(x)$ may be obtained as a partial case of the function $f_{y}$ for $z=x$ . Also, the ordinary derivative of odd-power $\frac{d}{dx}g_{y}$ evaluate in point $u\in\mathbb{R}$ may be obtained as a sum of partial derivatives of $f_{y}$ evaluate in point $(u,u)$ . We explain this further in the manuscript. One more important thing remains to conclude is to define partial derivative’s notation. More precisely, the following notation for partial derivatives is used across the manuscript and remains unchanged

Notation 1.3.

(Partial derivative.) Let be a function $\beta(x_{1},x_{2},\dots,x_{n})$ defined over the real space $\mathbb{R}^{n}$ . We denote partial derivative of the function $\beta$ with respect to $x_{i}$ as follows

\beta^{{}^{\prime}}_{x_{i}}=\lim_{\Delta x_{i}\to 0}\frac{\beta(x_{1},x_{2},\dots,x_{i}+\Delta x_{i},\dots,x_{n})-\beta(x_{1},x_{2},\dots,x_{n})}{\Delta x_{i}}

Partial derivative of the function $\beta_{x_{i}}$ with respect to $x_{i}$ evaluate in point $(y_{1},y_{2},\dots,y_{n})\in\mathbb{R}^{n}$ is denoted as follows

\beta^{{}^{\prime}}_{x_{i}}(y_{1},y_{2},\dots,y_{n})

Moreover, partial derivative $\beta^{{}^{\prime}}_{x_{i}}$ evaluate in point $(y_{1},y_{2},\dots,y_{n})\in\mathbb{R}^{n}$ plus partial derivative $\beta^{{}^{\prime}}_{x_{j}}$ evaluate in point $(y_{1},y_{2},\dots,y_{n})\in\mathbb{R}^{n}$ is equivalent to the sum of partial derivatives $\beta^{{}^{\prime}}_{x_{i}}+\beta^{{}^{\prime}}_{x_{j}}$ evaluate in point $(y_{1},y_{2},\dots,y_{n})\in\mathbb{R}^{n}$ and to be denoted as

\beta^{{}^{\prime}}_{x_{i}}(y_{1},y_{2},\dots,y_{n})+\beta^{{}^{\prime}}_{x_{j}}(y_{1},y_{2},\dots,y_{n})=[\beta^{{}^{\prime}}_{x_{i}}+\beta^{{}^{\prime}}_{x_{j}}](y_{1},y_{2},\dots,y_{n})

So that now we can switch our focus back to the functions $g_{y}$ and $f_{y}$ . Therefore, the following theorem in terms of partial derivatives reflects the relation between the ordinary derivative of odd-power function $g_{y}$ and function $f_{y}$

Theorem 1.4.

Let be a fixed point $v\in\mathbb{N}$ , then ordinary derivative $\frac{d}{dx}g_{v}(u)$ of the odd-power function $g_{v}(x)=x^{2v+1}$ evaluate in point $u\in\mathbb{R}$ equals to partial derivative $(f_{v})^{{}^{\prime}}_{x}(u,u)$ evaluate in point $(u,u)$ plus partial derivative $(f_{v})^{{}^{\prime}}_{z}(u,u)$ evaluate in point $(u,u)$

\frac{d}{dx}g_{v}(u)=(f_{v})^{{}^{\prime}}_{x}(u,u)+(f_{v})^{{}^{\prime}}_{z}(u,u)

(3)

In particular, it follows that for every pair $u\in\mathbb{R},v\in\mathbb{N}$ an identity holds

	$\displaystyle(2v+1)u^{2v}$	$\displaystyle=(f_{v})^{{}^{\prime}}_{x}(u,u)+(f_{v})^{{}^{\prime}}_{z}(u,u)$
		$\displaystyle=[(f_{v})^{{}^{\prime}}_{x}+(f_{v})^{{}^{\prime}}_{z}](u,u)$

that is also an ordinary derivative of odd-power function $t^{2v+1},\;v\in\mathbb{N},\;v=\mathrm{const}$ evaluate in point $u\in\mathbb{R}$ , therefore

	$\displaystyle\frac{d}{dt}t^{2v+1}(u)$	$\displaystyle=(f_{v})^{{}^{\prime}}_{x}(u,u)+(f_{v})^{{}^{\prime}}_{z}(u,u)$
		$\displaystyle=[(f_{v})^{{}^{\prime}}_{x}+(f_{v})^{{}^{\prime}}_{z}](u,u)$

To summarize and clarify all about the theorem 1.4, we provide a few examples that show an application of it.

Example 1.5.

Theorem 1.4 example for $x\in\mathbb{R},\;z\in\mathbb{R}$ and $y=1$ . Consider the explicit form of the function $f_{1}(x,z)$ , that is

f_{1}(x,z)=3xz-3z^{2}+3xz^{2}-2z^{3}

Therefore, the partial derivative $(f_{1})^{{}^{\prime}}_{x}$ with respect to $x$ equals to

(f_{1})^{{}^{\prime}}_{x}=\lim_{d\to 0}\frac{3dz+3dz^{2}}{d}=3z+3z^{2}

Consider the partial derivative $(f_{1})^{{}^{\prime}}_{z}$ with respect to $z$ , that is

	$\displaystyle(f_{1})^{{}^{\prime}}_{z}$	$\displaystyle=\lim_{d\to 0}\left[\frac{-3d^{2}-2d^{3}+3dx+3d^{2}x-6dz-6d^{2}z+6dxz-6dz^{2}}{d}\right]$
		$\displaystyle=\lim_{d\to 0}\left[-3d-2d^{2}+3x+3dx-6z-6dz+6xz-6z^{2}\right]$
		$\displaystyle=3x-6z+6xz-6z^{2}$

Summing up both partial derivatives $(f_{1})^{{}^{\prime}}_{x}$ and $(f_{1})^{{}^{\prime}}_{z}$ , we get

\displaystyle(f_{1})^{{}^{\prime}}_{x}+(f_{1})^{{}^{\prime}}_{z}=3x-3z+6xz-3z^{2}

Evaluating in point $(u,u)$ yields

\displaystyle\frac{d}{dt}t^{3}(u)=[(f_{1})^{{}^{\prime}}_{x}+(f_{1})^{{}^{\prime}}_{z}](u,u)=3u^{2}

That confirms the results of the theorem 1.4.

Example 1.6.

Theorem 1.4 example for $x\in\mathbb{R},\;z\in\mathbb{R}$ and $y=2$ . Consider the explicit form of the function $f_{2}(x,z)$ , that is

f_{2}(x,z)=5x^{2}z-15xz^{2}+15x^{2}z^{2}+10z^{3}-30xz^{3}+10x^{2}z^{3}+15z^{4}-15xz^{4}+6z^{5}

Therefore, the partial derivative $(f_{2})^{{}^{\prime}}_{x}$ with respect to $x$ equals to

	$\displaystyle(f_{2})^{{}^{\prime}}_{x}$	$\displaystyle=\lim_{d\to 0}\left[5dz+10xz-15z^{2}+15dz^{2}+30xz^{2}-30z^{3}+10dz^{3}+20xz^{3}-15z^{4}\right]$
		$\displaystyle=10xz-15z^{2}+30xz^{2}-30z^{3}+20xz^{3}-15z^{4}$

Consider the partial derivative $(f_{2})^{{}^{\prime}}_{z}$ with respect to $z$ , that is

\displaystyle(f_{2})^{{}^{\prime}}_{z}

\displaystyle=5x^{2}-30xz+30x^{2}z+30z^{2}-90xz^{2}+30x^{2}z^{2}+60z^{3}-60xz^{3}+30z^{4}

Summing up both partial derivatives $(f_{2})^{{}^{\prime}}_{x}$ and $(f_{2})^{{}^{\prime}}_{z}$ , we get

\displaystyle(f_{2})^{{}^{\prime}}_{x}+(f_{2})^{{}^{\prime}}_{z}

\displaystyle=5x^{2}-20xz+30x^{2}z+15z^{2}-60xz^{2}+30x^{2}z^{2}+30z^{3}-40xz^{3}+15z^{4}

Evaluate in point $(u,u)$ yields

\frac{d}{dt}t^{5}(u)=[(f_{2})^{{}^{\prime}}_{x}+(f_{2})^{{}^{\prime}}_{z}](u,u)=5u^{4}

That confirms the results of the theorem 1.4.

Example 1.7.

Theorem 1.4 example for $x\in\mathbb{R},\;z\in\mathbb{R}$ and $y=3$ . Consider the explicit form of the function $f_{3}(x,z)$ , that is

	$\displaystyle f_{3}(x,z)$	$\displaystyle=-7xz+14x^{2}z+7z^{2}-42xz^{2}+35x^{3}z^{2}+28z^{3}-140x^{2}z^{3}+70x^{3}z^{3}+175xz^{4}$
		$\displaystyle-210x^{2}z^{4}+35x^{3}z^{4}-70z^{5}+210xz^{5}-84x^{2}z^{5}-70z^{6}+70xz^{6}-20z^{7}$

Therefore, the partial derivative of $(f_{3})^{{}^{\prime}}_{x}$ with respect to $x$ equals to

	$\displaystyle(f_{3})^{{}^{\prime}}_{x}$	$\displaystyle=-7z+28xz-42z^{2}+105x^{2}z^{2}-280xz^{3}+210x^{2}z^{3}+175z^{4}-420xz^{4}$
		$\displaystyle+105x^{2}z^{4}+210z^{5}-168xz^{5}+70z^{6}$

Consider the partial derivative $(f_{3})^{{}^{\prime}}_{z}$ with respect to $z$ , that is

	$\displaystyle(f_{3})^{{}^{\prime}}_{z}$	$\displaystyle=-7x+14x^{2}+14z-84xz+70x^{3}z+84z^{2}-420x^{2}z^{2}+210x^{3}z^{2}+700xz^{3}$
		$\displaystyle-840x^{2}z^{3}+140x^{3}z^{3}-350z^{4}+1050xz^{4}-420x^{2}z^{4}-420z^{5}+420xz^{5}-140z^{6}$

Summing up both partial derivatives $(f_{3})^{{}^{\prime}}_{x}(x,z)$ and $(f_{3})^{{}^{\prime}}_{z}(x,z)$ , we get

	$\displaystyle(f_{3})^{{}^{\prime}}_{x}+(f_{3})^{{}^{\prime}}_{z}$	$\displaystyle=-7x+14x^{2}+7z-56xz+70x^{3}z+42z^{2}-315x^{2}z^{2}+210x^{3}z^{2}$
		$\displaystyle+420xz^{3}-630x^{2}z^{3}+140x^{3}z^{3}-175z^{4}+630xz^{4}-315x^{2}z^{4}-210z^{5}$
		$\displaystyle+252xz^{5}-70z^{6}$

Evaluate in point $(u,u)$ yields

\frac{d}{dt}t^{3}(u)=[(f_{3})^{{}^{\prime}}_{x}+(f_{3})^{{}^{\prime}}_{z}](u,u)=7u^{6}

That confirms the results of the theorem 1.4.

2. Conclusions

In this manuscript, we have reviewed an approach to get ordinary derivative of odd-power using an identity in partial derivatives of the function $f_{y}$ evaluate in fixed point $(u,u)\in\mathbb{R}^{2}.$ , that is described by the theorem 1.4. The main results of the manuscript can be validated using Mathematica programs available online at [4].

3. Verification of the results

As it is stated in conclusions, there is a possibility to validate the main results of this manuscript using Wolfram Mathematica. Therefore, a complete guide to validate the main results and formulae is attached as well. Mathematica package source file is available online under the folder mathematica, see [4]. The following expressions could be verified:

•

The function $f_{y}(x,z)$ for any constant argument $y\in\mathbb{N}$ using mathematica method f[x, y, z] e.g.

$\texttt{f[x, 1, z]}=3xz-3z^{2}+3xz^{2}-2z^{3}$
•

Partial derivative $(f_{y})^{{}^{\prime}}_{x}$ for any constant argument $y\in\mathbb{N}$ using mathematica method DerivativeFByX[x, y, z]

$\texttt{DerivativeFByX[x, 1, z]}=3z+3z^{2}$
•

Partial derivative $(f_{y})^{{}^{\prime}}_{z}$ for any constant argument $y\in\mathbb{N}$ using mathematica method DerivativeFByZ[x, y, z]

$\texttt{DerivativeFByZ[x, 1, z]}=3x-6z+6xz-6z^{2}$

•

Theorem 1.4 for any constant argument $y\in\mathbb{N}$

	$\displaystyle\texttt{DerivativeFByX[x, 1, z]}+\texttt{DerivativeFByZ[x, 1, z]}$	$\displaystyle=3x-3z+6xz-3z^{2}$
	$\displaystyle\texttt{DerivativeFByX[u, 1, u]}+\texttt{DerivativeFByZ[u, 1, u]}$	$\displaystyle=3u^{2}$

References

[1] Petro Kolosov. 106.37 An unusual identity for odd-powers. The Mathematical Gazette, 106(567):509–513, 2022. https://doi.org/10.1017/mag.2022.129.
[2] Petro Kolosov. On the link between Binomial Theorem and Discrete Convolution of Polynomials. arXiv preprint arXiv:1603.02468, 2016. https://arxiv.org/abs/1603.02468.
[3] Alan F. Beardon. Sums of powers of integers. The American mathematical monthly, 103(3):201–213, 1996.
[4] Kolosov, Petro. Another approach to get derivative of odd-power (Source files). Published electronically at https://github.com/kolosovpetro/AnotherApproachToGetDerivativeOfOddPower, 2022.

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