\section{The Notion of  Point-derivative}

Let $\cal{A}$ be a $K$-algebra. Recall that a $K$-linear map $d: \cal{A}\to\cal{A}$  is called  a {\it derivation} on $\cal{A}$ if $d(ab)=a d(b)+d(a) b$ for $a$, $b\in \cal{A}$. The axiomatic definition of  the point-derivative for  noncommutative algebras  is given in the following 

\medskip
\begin{definition}\label{def1.1} Let $\cal{A}$ be a $K$-algebra   and let $End(\cal{A})$ be the  set of all $K$-linear maps from $\cal{A}$ to $\cal{A}$. A $K$-linear map $\cal{D}: \cal{A}\to$  $End({\cal A})$  is called a 
{\bf point-derivation} on $\cal{A}$ if 
 \begin{description} 
\item[(i)] $\cal{D}_{\beta}:=\cal{D}(\beta)\in$ $End(\cal{A})$ is a derivation on $\cal{A}$ for all $\beta\in \cal{A}$,
\item[(ii)] ${\cal D}_z=z\,{\cal D}_1$ for all $z$ in the center of ${\cal A}$,
\end{description}
where $1$ is the identity of the algebra ${\cal A}$. For $f\in {\cal A}$ and $\beta\in \cal{A}$, 
${\cal D}_{\beta}(f)\in  {\cal A}$ is called the {\bf point-derivative depending on $\beta$} of $f$ or  {\bf $\beta$-derivative} of $f$  .
\end{definition}

\medskip
If  $\cal{A}$ is a commutative algebra, then  a  derivation $d$ on ${\cal A}$ produces naturally a  point-derivation ${\cal D}$ on 
${\cal A}$ as follows:
$$ {\cal D}(\beta):=\beta d\quad\mbox{for $\beta\in {\cal A}$},$$
where $\beta d\in End({\cal A})$ is defined by $(\beta d)(a): =\beta\cdot d(a)$ for $a\in {\cal A}$.

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We now explain how to introduce  point-derivative in noncommutative  formal power series. Let $A$ be the alphabet set which has a distinguished letter, denoted by $x$ and called the {\it variable},  and an infinite supply of other letters $a$, $b$, \dots called 
{\it constants}.  A {\it word} $w$ is an element of the free monoid  $M$ generated by $A$, and  the {\it degree}
of a word $w$  is the number of occurrences of the letter $x$ in $w$.
The {\it empty word}, which is denoted by $1$, is the identity element of free monoid $M$. Following \cite{RSS}, we  use   
$K\{\{a, b, \dots, x\}\}$ to denote the algebra of noncommutative formal power series in a single 
variable $x$ over the field $K$.  

\medskip
Let $\beta\in K\{\{a, b, \dots, x\}\}$ be a fixed formal power series.  For a given word 
$w=c_1x^{i_1}c_2x^{i_2}\cdots c_nx^{i_n}c_{n+1}\in M$, where $c_1$, $c_2$, $\cdots$, $c_n$, $c_{n+1}$ are the words of degree $0$ and $i_1$, $i_2$, $\cdots$, $i_n$ are nonnegative integers, we define a map  $\displaystyle\frac{d}{d_{\beta}x}$ from $M$ to $K\{\{a, b, \dots, x\}\}$ as follows:
$$
\displaystyle\frac{d}{d_{\beta}x}(w):=0\quad\mbox{if $i_1=i_2=\cdots =i_n=0$}
$$
and
\begin{eqnarray*}
\displaystyle\frac{d}{d_{\beta}x}(w):&=&\displaystyle\sum_{k=1}^nc_1x^{i_1}\cdots c_{k-1}x^{i_{k-1}}
c_k\cdot\displaystyle\frac{d}{d_{\beta}x}(x^{i_k})\cdot  c_{k+1}x^{i_{k+1}}\cdots c_nx^{i_n}c_{n+1}\\
&&\mbox{if $i_1i_2\cdots i_n> 0$,}
\end{eqnarray*}
where $c_1x^{i_1}\cdots c_{k-1}x^{i_{k-1}}:=1$ for $k=1$,  $c_{k+1}x^{i_{k+1}}\cdots c_nx^{i_n}c_{n+1}:=1$ for $k=n$ and 
 $\displaystyle\frac{d}{d_{\beta}x}(x^{i_k})$ is defined  by
$$
\displaystyle\frac{d}{d_{\beta}x}(x^{i_k})=\left\{\begin{array}{ll}\beta&
\mbox{if $i_k=1$}\\
x^{i_k-1}\beta+x^{i_k-2}\beta x+\cdots +x\beta x^{i_k-2}+\beta x^{i_k-1}&\mbox{if $i_k> 1$}\end{array}\right..
$$
After extending the map $\displaystyle\frac{d}{d_{\beta}x}:M\to K\{\{a, b, \dots, x\}\}$ linearly and continuously, we get a 
$K$-linear map $\displaystyle\frac{d}{d_{\beta}x}: K\{\{a, b, \dots, x\}\}\to K\{\{a, b, \dots, x\}\}$, which is called the
{\bf point-derivative operator depending on $\beta$}. Clearly, The map: 
$\beta\to \displaystyle\frac{d}{d_{\beta}x}$ is a point-derivation on $K\{\{a, b, \dots, x\}\}$.   
The  {\bf $\beta$-derivative}  $\displaystyle\frac{d}{d_{\beta}x}\big(f(x)\big)$
  of a formal power series $f(x)\in K\{\{a, b, \dots, x\}\}$ is also denoted by  
$\displaystyle\frac{df}{d_{\beta}x}$  or $f'_{\beta}(x)$.
 Note that  the $1$--derivative $\displaystyle\frac{df}{d_{1}x}=f'_{1}(x)$ of the  formal power series $f(x)$ is the Hausdorff derivative $H<f>$ of $f(x)$.

\medskip
Like the Hausdorff derivative and cyclic derivative, we have the following characterization of the  point-derivative operator 
 $\displaystyle\frac{d}{d_{\beta}x}$.

\begin{proposition}\label{pr1.1} Let  $\beta\in K\{\{a, b, \dots, x\}\}$ be a fixed formal power series. The point--derivative operator 
 $\displaystyle\frac{d}{d_{\beta}x}:  K\{\{a, b, \dots, x\}\}\to K\{\{a, b, \dots, x\}\}$ is the unique $K$-linear continuous operator on 
$ K\{\{a, b, \dots, x\}\}$ satisfying 
\begin{description} 
\item[(i)] $\displaystyle\frac{d}{d_{\beta}x}(a)=0$ for any constant $a$,
\item[(ii)] $\displaystyle\frac{d}{d_{\beta}x}(x)=\beta$,
\item[(iii)] $\displaystyle\frac{d}{d_{\beta}x}(fg)=\displaystyle\frac{d}{d_{\beta}x}(f)\cdot g+f\cdot\displaystyle\frac{d}{d_{\beta}x}(g)$ for  $f$,  $g\in  K\{\{a, b, \dots, x\}\}$.
\end{description}
\end{proposition}

\bigskip
\section{Partial Point-derivatives}

For convenience, we use  $K\{\{a, b, \dots, || x, y\}\}$ to denote the algebra of noncommutative formal power series in two  noncommutative  variables $x$ and $y$.  Let   $f(x, y)\in K\{\{a, b, \dots, || x, y\}\}$  and 
$\beta=\beta(x, y)\in K\{\{a, b, \dots, || x, y\}\}$ be   formal power series in two  noncommutative  variables $x$ and $y$. 
 If we keep $y$ constant, then  $f(x, y)$ and $\beta=\beta(x, y)$ are  formal power series of $x$ and its  $\beta$-derivative  with respect to the variable $x$ is denoted by $ \displaystyle\frac{\partial f(x, y)}{\partial_{\beta} x}$ or  $ \displaystyle\frac{\partial f}{\partial_{\beta} x}$, which is called 
the {\bf partial  $\beta$-derivative of $f(x, y)$  with respect to the variable $x$}.  
 If we keep $x$ constant, then $f(x, y)$  and $\beta=\beta(x, y)$ are  formal power series of $y$ and its  $\beta$-derivative  with respect to the variable $y$ is denoted by $ \displaystyle\frac{\partial f(x, y)}{\partial_{\beta} y}$ or  $ \displaystyle\frac{\partial f}{\partial_{\beta} y}$, which is called 
the {\bf partial  $\beta$-derivative of $f(x, y)$  with respect to the variable $y$}.  

\medskip
Let $\beta=\beta(x, y)$, $\gamma=\gamma (x, y)$ and  $f=f(x, y)$ be  formal 
power series in   noncommutative  variables $x$ and $y$.    We define  {\bf the second partial 
 $( _{\gamma}y,  \, _{\beta}x)$-derivative 
$\displaystyle\frac{\partial^2 f}{\partial_{\gamma} y\partial_{\beta} x}$ of $f(x, y)$} and
 {\bf the second partial 
 $( _{\beta}x,  \, _{\gamma}y)$-derivative 
$\displaystyle\frac{\partial^2 f}{\partial_{\beta} x\partial_{\gamma} y}$ of $f(x, y)$} by
$$
\displaystyle\frac{\partial^2 f}{\partial_{\gamma} y\partial_{\beta} x}:=
\displaystyle\frac{\partial }{\partial_{\gamma} y}\left(\displaystyle\frac{\partial f}{\partial_{\beta} x}\right)
\quad\mbox{and}\quad
\displaystyle\frac{\partial^2 f}{\partial_{\beta} x\partial_{\gamma} y}:=
\displaystyle\frac{\partial }{\partial_{\beta} x}\left(\displaystyle\frac{\partial f}{\partial_{\gamma} y}\right).
$$

\medskip
The following proposition extends  multi-variable chain rule 
and Clairaut's Theorem to  partial point-derivatives for formal power series  containing $x$ and $y$.  

\medskip
\begin{proposition}\label{pr2.3} If $f=f(x, y)$,  $\beta=\beta(x, y)$, $\gamma=\gamma (x, y)$, 
$u=u(x, y)$ and $v=v(x, y)$ are  formal power series in two  noncommutative  variables $x$ and $y$, then
$$
\displaystyle\frac{\partial \big(f(u, v)\big)}{\partial_{\beta} x}=
\displaystyle\frac{\partial \big(f(u, v)\big)}{\partial_{\frac{\partial u}{\partial_{\beta} x}} u}+
\displaystyle\frac{\partial \big(f(u, v)\big)}{\partial_{\frac{\partial v}{\partial_{\beta} x}} v}.
$$
and
$$
\displaystyle\frac{\partial^2 f}{\partial_{\gamma} y\partial_{\beta} x}-
\displaystyle\frac{\partial^2 f}{\partial_{\beta} x\partial_{\gamma} y}=
\displaystyle\frac{\partial f}{\partial_{\big(\frac{\partial \beta}{\partial_{\gamma} y}\big)} x}-
\displaystyle\frac{\partial f}{\partial_{\big(\frac{\partial \gamma}{\partial_{\beta} x}\big)} y}.
$$
\end{proposition}

\medskip
The proof of the proposition above follows from a direct computation.

\bigskip
By \cite{CKW}, the ordinary partial derivatives is a useful tool in the study of commutative arithmetic
circuits. Hence, one problem, which is worthy to study it in future, is to determine if  the
partial point-derivatives can be used to study non-commutative arithmetic circuits.


\bigskip
