The Directional Derivative in General Quantum Calculus

Abstract

In this paper, we define the $\beta$-partial derivative as well as the $\beta$-directional derivative of a multi-variable function based on the $\beta$-difference operator, $D_{\beta }$, which is defined by $D_{\beta }f\left(t\right)=\left(f\left(\beta \right(t\left)\right)-f\left(t\right)\right)/\left(\beta \left(t\right)-t\right)$, where $\beta$ is a strictly increasing continuous function. Some properties are proved. Furthermore, the $\beta$-gradient vector and the $\beta$-gradient directional derivative of a multi-variable function are introduced. Finally, we deduce the Hahn-partial and the Hahn-directional derivatives associated with the Hahn difference operator.

1. Introduction and Preliminaries

Calculus without limits is known as quantum calculus. Roughly speaking, quantum calculus is an approach to deal with non-differentiable functions. Similar to fractional calculus, it is related to the classical derivative operator. Although there is a common bond, they are very different branches of mathematics. The fractional branch consider functions that may have no derivative of first order but have fractional derivatives of orders less than one [1,2]; however, the quantum branch is particularly useful for modeling physical systems (see, for instance, [3,4,5,6]).

Quantum calculus began with FH Jackson in the early twentieth century, but earlier, this type of calculus had been worked out by Euler and Jacobi. For the reader, we recommend the book by Kac and Cheung [7] for many fundamental features of quantum calculus. Quantum difference operators and their calculi have attracted many researchers in recent years due to their applications in different branches of mathematics as well as physics, such as orthogonal polynomials, basic hypergeometric series, variational calculus, combinatorics, model physical and economical systems, quantum mechanics, and black holes (see, for example, [8,9,10,11,12,13,14,15]). As it usually happens in mathematics, the corresponding inverse operators (integral operators) are important directly related topics. There are plenty of quantum operators and associated calculi, such as forward, Jackson, Hahn, power quantum, and $p,q$-calculi (see [7,16,17]). Moreover, a symmetric quantum calculus was introduced in [18,19], based on the symmetric quantum difference operator, which is defined by

$$D_{\alpha ,\gamma }f\left(t\right)=\frac{f(t+\alpha )-f(t-\gamma )}{\alpha +\gamma },\alpha ,\gamma \in [0,\infty ).$$

(1)

Note that in the case of $\gamma =0$ in (1), we obtain the forward difference operator. In 2015, Hamza et al. [20] defined the general quantum difference operator denoted by $D_{\beta }$ as

$$D_{\beta }f\left(t\right)=\left\{\begin{array}{c}{\displaystyle \frac{f\left(\beta \right(t\left)\right)-f\left(t\right)}{\beta \left(t\right)-t},t\ne \beta \left(t\right),}\hfill \\ \\ f^{\prime }\left(t\right),t=\beta \left(t\right),\hfill \end{array}\right.$$

(2)

where $\beta \left(t\right)$ is a continuous and strictly increasing function defined on an interval $I\subset \mathbb{R}$ such that $\beta \left(t\right)\in I$ whenever $t\in I$. The function $f:I\to \mathbb{R}$ is said to be $\beta$-differentiable if $f^{\prime }\left(t\right)$ exists at the fixed points of the function $\beta$. Moreover, in [20], the authors started the calculus associated with $D_{\beta }$ when the function $\beta$ has only a unique fixed point $s_0\in I$ and satisfies the condition $(t-s_0)(\beta \left(t\right)-t)\le 0$ for all $t\in I$. This operator produces the Jackson q-difference operator when $\beta \left(t\right)=qt$, $t\in I$, $q\in (0,1)$ [7]. Additionally, $D_{\beta }$ produces the Hahn difference operator, when $\beta \left(t\right)=qt+\omega$, $q\in (0,1)$, and $\omega >0$, which is defined in [21] by

$$D_{q,\omega }f\left(t\right)=\left\{\begin{array}{c}{\displaystyle \frac{f(qt+\omega )-f\left(t\right)}{t(q-1)+\omega },t\ne \frac{\omega }{1-q},}\hfill \\ \\ f^{\prime }\left(t\right),t=\frac{\omega }{1-q}.\hfill \end{array}\right.$$

(3)

Furthermore, $D_{\beta }$ yields the forward difference operator when $\beta \left(t\right)=t+\omega$, $\omega >0$ [7]. In the case of $\beta \left(t\right)=q{t}^n$, n is an odd number, $q\in (0,1)$, we obtain the power quantum operator (see [17]). In [22], many types of the function $\beta$ were discussed according to the number of its fixed points located in the interval I as a basis for different associated calculi. The operator ${\displaystyle D_{\beta }}$, similar to many topics in mathematics, presents both symmetric and asymmetric characteristics, these aspects being visible in the operator itself and in the identities that the reader can find in the corresponding results of Section 2.

The $\beta$-product and the $\beta$-quotient rules are given in [20] by

$$\begin{array}{cc}\hfill D_{\beta }\left(fg\right)\left(t\right)=& D_{\beta }\left(f\left(t\right)\right)g\left(t\right)+f\left(\beta \left(t\right)\right)D_{\beta }g\left(t\right)\hfill \\ & =D_{\beta }\left(f\left(t\right)\right)g\left(\beta \left(t\right)\right)+f\left(t\right)D_{\beta }g\left(t\right)\hfill \end{array}$$

and

$$D_{\beta }\left(\frac{f}{g}\right)\left(t\right)=\frac{D_{\beta }\left(f\left(t\right)\right)g\left(t\right)-f\left(t\right)D_{\beta }g\left(t\right)}{g\left(t\right)g\left(\beta \right(t\left)\right)},g\left(t\right)g\left(\beta \left(t\right)\right)\ne 0,$$

respectively.

Proposition 1.

Let $f:I\to \mathbb{R}$ be a given function and $g:I\to I$ a function such that $g\left(\beta \left(t\right)\right)\ne g\left(t\right)$. Then, whenever $\beta \left(t\right)\ne t$,

$$\frac{\left(D_{\beta }f\circ g\right)\left(t\right)}{D_{\beta }g\left(t\right)}=\frac{f\left(g\left(\beta \left(t\right)\right)\right)-f\left(g\left(t\right)\right)}{g\left(\beta \left(t\right)\right)-g\left(t\right)},$$

where ∘ is the usual composition of functions.

Proof. 

The proof is trivial since, for $\beta \left(t\right)\ne t$, we have

$$\left(D_{\beta }f\circ g\right)\left(t\right)=\frac{\left(f\circ g\right)\left(\beta \left(t\right)\right)-\left(f\circ g\right)\left(t\right)}{\beta \left(t\right)-t}=\frac{f\left(g\left(\beta \left(t\right)\right)\right)-f\left(g\left(t\right)\right)}{\beta \left(t\right)-t},$$

therefore

$$\left(D_{\beta }f\circ g\right)\left(t\right)=\frac{f\left(g\left(\beta \left(t\right)\right)\right)-f\left(g\left(t\right)\right)}{g\left(\beta \left(t\right)\right)-g\left(t\right)}\frac{g\left(\beta \left(t\right)\right)-g\left(t\right)}{\beta \left(t\right)-t}.$$

Hence,

$$\frac{f\left(g\left(\beta \left(t\right)\right)\right)-f\left(g\left(t\right)\right)}{g\left(\beta \left(t\right)\right)-g\left(t\right)}=\frac{\left(D_{\beta }f\circ g\right)\left(t\right)}{\left(D_{\beta }g\right)\left(t\right)}.$$

□

The $\beta$-integral of a function $f:I\to \mathbb{R}$ is defined in [20] by

$${\int }_a^{b}f\left(t\right)d_{\beta }t={\int }_{s_0}^{b}f\left(t\right)d_{\beta }t-{\int }_{s_0}^{a}f\left(t\right)d_{\beta }t,a,b\in I,$$

where

$$F\left(c\right)={\int }_{s_0}^{c}f\left(t\right)d_{\beta }t=\sum _{k=0}^{\infty }\left({\beta }^k\left(c\right)-{\beta }^{k+1}\left(c\right)\right)f\left({\beta }^k\left(c\right)\right),c\in I,$$

when the series on the right hand side converges. Some integral inequalities based on $D_{\beta }$ were introduced in [23]. Moreover, the $\beta$-Laplace transform and the $\beta$-convolution theorem were given in [24,25]. For more details about the $\beta$-calculus associated with $D_{\beta }$, see [26,27,28,29]. The benefit of using the $\beta$-calculus is that, from its results, we can deduce the other quantum calculi results and, this way, exhibit the properties of quantum calculus in a unified way. It also gives the possibility to study other particular quantum operators distinct from the known ones.

In this paper, we define the $\beta$-partial derivative of n-variable functions, and we prove some properties. Furthermore, we introduce the $\beta$-directional and the $\beta$-gradient derivatives. Moreover, from the general quantum calculus, we deduce these results in the quantum calculus associated with the Hahn-difference operator (3).

2. Main Results

2.1. On the $\beta$-Partial Derivative

Let f be a function of two variables, and I be a non-empty closed subset of the real numbers $\mathbb{R}$. In [29], the $\beta$-partial derivatives of the function $f(x,y)$ with respect to x and y are defined by

$$\frac{\partial f}{{\partial }_{\beta }x}\equiv \frac{\partial f(x,y)}{{\partial }_{\beta }x}\equiv \frac{\partial f}{{\partial }_{\beta }x}(x,y):=\frac{f\left(\beta \right(x),y)-f(x,y)}{\beta \left(x\right)-x}$$

(4)

and

$$\frac{\partial f}{{\partial }_{\beta }y}\equiv \frac{\partial f(x,y)}{{\partial }_{\beta }y}\equiv \frac{\partial f}{{\partial }_{\beta }y}(x,y):=\frac{f(x,\beta (y\left)\right)-f(x,y)}{\beta \left(y\right)-y},$$

(5)

respectively, $x,y\in I$, $x\ne s_0$, and $y\ne s_0$. We note, according to the general quantum operator (2), that these definitions become the classical partial derivatives $\frac{\partial f\left(s_0,y\right)}{\partial x}$ and $\frac{\partial f(x,s_0)}{\partial y}$ in the cases where $x=s_0$ and $y=s_0$, respectively, whenever they exist, where $s_0\in I$ is a fixed point of the function $\beta$.

The product rule and the quotient rule of the $\beta$-partial derivative with respect to $x$, $x\ne s_0$ are given, respectively, by

$$\begin{array}{cc}\hfill \frac{\partial }{{\partial }_{\beta }x}\left(fg\right)(x,y)& =\left[\frac{\partial }{{\partial }_{\beta }x}f(x,y)\right]g(x,y)+f(\beta \left(x\right),y)\left[\frac{\partial }{{\partial }_{\beta }x}g(x,y)\right]\hfill \\ & =\left[\frac{\partial }{{\partial }_{\beta }x}f(x,y)\right]g\left(\beta \left(x\right),y\right)+f(x,y)\left[\frac{\partial }{{\partial }_{\beta }x}g(x,y)\right],\hfill \end{array}$$

where $\left(fg\right)(x,y)=f(x,y)g(x,y)$, and

$$\frac{\partial }{{\partial }_{\beta }x}(f/g)(x,y)=\frac{\left[\frac{\partial }{{\partial }_{\beta }x}f(x,y)\right]g(x,y)-f(x,y)\left[\frac{\partial }{{\partial }_{\beta }x}g(x,y)\right]}{g(x,y)g\left(\beta \right(x),y)},$$

where $g(x,y)g\left(\beta \right(x),y)\ne 0$ [25].

The case with respect to $y$ is similar.

Now, we define the $\beta$-partial derivative of a multi-variable function.

Definition 1.

Let $f(x_1,x_2,\dots ,x_n)$ be a real multi-variable function, where $x_1,x_2,\dots ,x_n\in I$. Then, the β-partial derivative of the function $f$ with respect to $x_i$ is defined whenever $x_i\ne s_0$ by

$$\frac{\partial f(x_1,x_2,\dots ,x_i,\dots ,x_n)}{{\partial }_{\beta }{x}_i}=\frac{f(x_1,x_2,\dots ,\beta \left(x_i\right),\dots x_n)-f(x_1,x_2,\dots ,x_i,\dots ,x_n)}{\beta \left(x_i\right)-x_i},$$

(6)

for$i=1,2,\dots ,n$.

We remark that one obtains the classical partial derivative $\frac{\partial f(x_1,x_2,\dots ,x_i,\dots ,x_n)}{\partial x_i}$ if $x_i=s_0$ is a fixed point of the function $\beta$. It is easy to prove that the $\beta$-partial derivative is a linear operator.

Proposition 2.

Let $f$ and $g$ be two real functions of n-variables, with each one of these variables defined on $I$ and $a,b\in \mathbb{R}$. Then,

$$\frac{\partial \left(af+bg\right)}{{\partial }_{\beta }{x}_i}\equiv \frac{\partial }{{\partial }_{\beta }{x}_i}\left(af+bg\right)=a\frac{\partial f}{{\partial }_{\beta }{x}_i}+b\frac{\partial g}{{\partial }_{\beta }{x}_i}.$$

Proof. 

The proof is straightforward using the definition of the $\beta$-partial derivative. □

Proposition 3.

Let$f$ and $g$ be two real functions of n-variables $x_i\in I,i=1,2,\dots ,n$. Then, for each $i=1,2,\dots ,n$ ,

$$\begin{array}{c}\frac{\partial }{{\partial }_{\beta }{x}_i}\left(fg\right)(x_1,\dots ,x_i,\dots ,x_n)=\left[\frac{\partial }{{\partial }_{\beta }{x}_i}f(x_1,\dots ,x_i,\dots ,x_n)\right]g(x_1,\dots ,x_i,\dots ,x_n)\hfill \\ +f(x_1,\dots ,x_{i-1},\beta \left(x_i\right),x_{i+1},\dots ,x_n)\left[\frac{\partial }{{\partial }_{\beta }{x}_i}g(x_1,\dots ,x_i,\dots ,x_n)\right]\hfill \\ =\left[\frac{\partial }{{\partial }_{\beta }{x}_i}f(x_1,\dots ,x_n)\right]g\left(x_1,\dots ,\beta \left(x_i\right),\dots ,x_n\right)+f(x_1,\dots ,x_n)\left[\frac{\partial }{{\partial }_{\beta }{x}_i}g(x_1,\dots ,x_n)\right]\hfill \end{array}$$

where $\left(fg\right)(x_1,x_2,\dots ,x_n)=f(x_1,x_2,\dots ,x_n)g(x_1,x_2,\dots ,x_n)$ and

$$\begin{array}{c}\frac{\partial }{{\partial }_{\beta }{x}_i}(f/g)(x_1,\dots ,x_i,\dots ,x_n)=\hfill \\ {\displaystyle \frac{\left[\frac{\partial }{{\partial }_{\beta }{x}_i}f(x_1,\dots ,x_n)\right]g(x_1,\dots ,x_n)-f(x_1,\dots ,x_n)\left[\frac{\partial }{{\partial }_{\beta }{x}_i}g(x_1,\dots ,x_n)\right]}{g\left(x_1,\dots ,x_n\left)g\right(x_1,\dots ,\beta \left(x_i\right),\dots ,x_n\right)},}\hfill \end{array}$$

where $g(x_1,x_2,\dots ,x_i,\dots ,x_n)g(x_1,x_2,\dots ,\beta \left(x_i\right),\dots ,x_n)\ne 0$.

Proof. 

Using the definition of the $\beta$-partial derivative with respect to the variable $x_i$ (6), we have

$$\begin{array}{c}\frac{\partial }{{\partial }_{\beta }{x}_i}\left(fg\right)(x_1,\dots ,x_n)=\frac{\left(fg\right)(x_1,\dots ,\beta \left(x_i\right),\dots ,x_n)-\left(fg\right)(x_1,\dots ,x_i,\dots ,x_n)}{\beta \left(x_i\right)-x_i}\hfill \\ =\frac{f\left(x_1,\dots ,\beta \left(x_i\right),\dots ,x_n\right)g\left(x_1,\dots ,\beta \left(x_i\right),\dots ,x_n\right)-f\left(x_1,\dots ,x_i,\dots ,x_n\right)g\left(x_1,\dots ,x_i,\dots ,x_n\right)}{\beta \left(x_i\right)-x_i}\hfill \\ =\frac{f\left(x_1,\dots ,\beta \left(x_i\right),\dots ,x_n\right)g\left(x_1,\dots ,\beta \left(x_i\right),\dots ,x_n\right)-f\left(x_1,\dots ,\beta \left(x_i\right),\dots ,x_n\right)g\left(x_1,\dots ,x_i,\dots ,x_n\right)}{\beta \left(x_i\right)-x_i}\hfill \\ +\frac{f\left(x_1,\dots ,\beta \left(x_i\right),\dots ,x_n\right)g\left(x_1,\dots ,x_i,\dots ,x_n\right)-f\left(x_1,\dots ,x_i,\dots ,x_n\right)g\left(x_1,\dots ,x_i,\dots ,x_n\right)}{\beta \left(x_i\right)-x_i}\hfill \\ =f\left(x_1,\dots ,\beta \left(x_i\right),\dots ,x_n\right)\left[\frac{\partial }{{\partial }_{\beta }{x}_i}g\left(x_1,\dots ,x_i,\dots ,x_n\right)\right]\hfill \\ +\left[\frac{\partial }{{\partial }_{\beta }{x}_i}f\left(x_1,\dots ,x_i,\dots ,x_n\right)\right]g\left(x_1,\dots ,x_i,\dots ,x_n\right).\hfill \end{array}$$

Similarly, we can prove the quotient rule. □

Proposition 1 motivates the following two definitions (notations): when $f(x,y)$ and $x=x\left(t\right)$, $y=y\left(t\right)$ are composed for $t\in I$, that is, when we are dealing with the operation of function composition and we want to consider its $\beta$-partial derivative with respect to the first component but acting on the variable $t$, to distinguish from the notations (4) and (5), we write

$$\frac{\partial f(x,y)}{{\partial }_{{\beta }_t}x}\equiv \frac{\partial f(x,y)}{{\partial }_{{\beta }_t}x}\left(t\right):=\frac{f\left(x\left(\beta \right(t\left)\right),y\left(t\right)\right)-f\left(x\left(t\right),y\left(t\right)\right)}{x\left(\beta \right(t\left)\right)-x\left(t\right)}$$

(7)

and

$$\frac{\partial f(x,y)}{{\partial }_{{\beta }_t}y}\equiv \frac{\partial f(x,y)}{{\partial }_{{\beta }_t}y}\left(t\right):=\frac{f\left(x\left(t\right),y\left(\beta \right(t\left)\right)\right)-f\left(x\left(t\right),y\left(t\right)\right)}{y\left(\beta \right(t\left)\right)-y\left(t\right)}.$$

(8)

Remark 1.

We observe that these definitions correspond to a partial version notation of Proposition 1. Note, also, that these definitions turn into the partial derivatives $\frac{\partial f\left(x\left(s_0\right),y\left(s_0\right)\right)}{\partial x}$ and $\frac{\partial f\left(x\left(s_0\right),y\left(s_0\right)\right)}{\partial y}$ for any fixed point $t=s_0$ of the function $\beta$.

Lemma 1.

Let $f:I\times I\to \mathbb{R}$ be a real function of two variables $(x,y)$. If $x=x\left(t\right)$ and $y=y\left(t\right)$ are two real functions with domain and range on I, then $f(x,y)\equiv f\left(x\left(t\right),y\left(t\right)\right)$ can be regarded as a real function of $t$ and

$$\begin{array}{ccc}{\displaystyle D_{\beta }f\left(x\left(t\right),y\left(t\right)\right)}\hfill & =\hfill & \frac{\partial f\left(x\left(t\right),y\left(\beta \right(t\left)\right)\right)}{{\partial }_{{\beta }_t}x\left(t\right)}{D}_{\beta }x\left(t\right)+\frac{\partial f\left(x\left(t\right),y\left(t\right)\right)}{{\partial }_{{\beta }_t}y\left(t\right)}{D}_{\beta }y\left(t\right)\hfill \\ {\displaystyle }\hfill & =\hfill & \frac{\partial f\left(x\left(t\right),y\left(t\right)\right)}{{\partial }_{{\beta }_t}x\left(t\right)}{D}_{\beta }x\left(t\right)+\frac{\partial f\left(x\left(\beta \right(t\left)\right),y\left(t\right)\right)}{{\partial }_{{\beta }_t}y\left(t\right)}{D}_{\beta }y\left(t\right).\hfill \end{array}$$

(9)

Proof. 

Let $z\left(t\right):=f\left(x\left(t\right),y\left(t\right)\right)\equiv f\left(x,y\right)\left(t\right)$. Then, the $\beta$-derivative of $z\left(t\right)$, $t\ne s_0$, is given by

$$\begin{array}{c}{\displaystyle D_{\beta }z\left(t\right)=\frac{z\left(\beta \right(t\left)\right)-z\left(t\right)}{\beta \left(t\right)-t}=\frac{f\left(x\left(\beta \right(t\left)\right),y\left(\beta \right(t\left)\right)\right)-f\left(x\left(t\right),y\left(t\right)\right)}{\beta \left(t\right)-t}}\hfill \\ {\displaystyle =\frac{f\left(x\left(\beta \right(t\left)\right),y\left(\beta \right(t\left)\right)\right)-f\left(x\left(t\right),y\left(\beta \right(t\left)\right)\right)+f\left(x\left(t\right),y\left(\beta \right(t\left)\right)\right)-f\left(x\left(t\right),y\left(t\right)\right)}{\beta \left(t\right)-t}}\hfill \\ {\displaystyle =\frac{f\left(x\left(\beta \right(t\left)\right),y\left(\beta \right(t\left)\right)\right)-f\left(x\left(t\right),y\left(\beta \right(t\left)\right)\right)}{\beta \left(t\right)-t}+\frac{f\left(x\left(t\right),y\left(\beta \right(t\left)\right)\right)-f\left(x\left(t\right),y\left(t\right)\right)}{\beta \left(t\right)-t}.}\hfill \end{array}$$

(10)

Considering, from (10), the notations

$$S_1=\frac{f\left(x\left(\beta \right(t\left)\right),y\left(\beta \right(t\left)\right)\right)-f\left(x\left(t\right),y\left(\beta \right(t\left)\right)\right)}{\beta \left(t\right)-t}$$

and

$$S_2=\frac{f\left(x\left(t\right),y\left(\beta \right(t\left)\right)\right)-f\left(x\left(t\right),y\left(t\right)\right)}{\beta \left(t\right)-t},$$

then, using definition (7), we have

$$\begin{array}{cc}\hfill S_1& =\frac{f\left(x\left(\beta \right(t\left)\right),y\left(\beta \right(t\left)\right)\right)-f\left(x\left(t\right),y\left(\beta \right(t\left)\right)\right)}{x\left(\beta \left(t\right)\right)-x\left(t\right)}\frac{x\left(\beta \left(t\right)\right)-x\left(t\right)}{\beta \left(t\right)-t}\hfill \\ & =\frac{\partial f\left(x\left(t\right),y\left(\beta \right(t\left)\right)\right)}{{\partial }_{{\beta }_t}x\left(t\right)}{D}_{\beta }x\left(t\right)\hfill \end{array}$$

and, by the use of definition (8),

$$\begin{array}{cc}\hfill S_2& =\frac{f\left(x\left(t\right),y\left(\beta \right(t\left)\right)\right)-f\left(x\left(t\right),y\left(t\right)\right)}{y\left(\beta \left(t\right)\right)-y\left(t\right)}\frac{y\left(\beta \left(t\right)\right)-y\left(t\right)}{\beta \left(t\right)-t}\hfill \\ & =\frac{\partial f\left(x\left(t\right),y\left(t\right)\right)}{{\partial }_{{\beta }_t}y\left(t\right)}{D}_{\beta }y\left(t\right).\hfill \end{array}$$

By inserting these last expressions for $S_1$ and $S_2$ into (10), we obtain the first identity of (9).

To obtain the second identity, we proceed in a similar manner. □

Remark 2.

When the variable under consideration is clear, we may naturally simplify the notations in (9) and write, for instance, for the first identity,

$$D_{\beta }f\left(x,y\right)=\frac{\partial f\left(x,y\circ \beta \right)}{{\partial }_{{\beta }_t}x}{D}_{\beta }x+\frac{\partial f(x,y)}{{\partial }_{{\beta }_t}y}{D}_{\beta }y,$$

where the symbol ∘ denotes the classical composition of functions.

Note that if we take $\beta \left(x\right)=qx$, $q\in (0,1)$, we obtain the q-partial derivative in the q-calculus. See [30].

Of course, using a similar technique, one can generalize Lemma 1 to higher dimensions. One obtains the result that follows.

Lemma 2.

Let $f:I^n\to \mathbb{R}$ be a function of $n$ real variables $(x_1,x_2,\dots ,x_n)$.

If $x_i:I\to I$, for each $i=1,2,\dots ,n$, can be considered a function of $t$, then $f\left(x_1,x_2,\dots ,x_n\right)\equiv f\left(x_1\left(t\right),x_2\left(t\right),\dots ,x_n\left(t\right)\right)$ can be regarded as a real function of $t$ and

$$\begin{array}{c}{D}_{\beta }f\left(x_1\left(t\right),x_2\left(t\right),\dots ,x_n\left(t\right)\right)=\hfill \\ \frac{\partial f\left(x_1\left(t\right),x_2\left(\beta \left(t\right)\right),\dots ,x_n\left(\beta \left(t\right)\right)\right)}{{\partial }_{{\beta }_t}{x}_1\left(t\right)}{D}_{\beta }{x}_1\left(t\right)+\frac{\partial f\left(x_1\left(t\right),x_2\left(t\right),x_3\left(\beta \left(t\right)\right),\dots ,x_n\left(\beta \left(t\right)\right)\right)}{{\partial }_{{\beta }_t}{x}_2\left(t\right)}{D}_{\beta }{x}_2\left(t\right)\hfill \\ +\frac{\partial f\left(x_1\left(t\right),x_2\left(t\right),x_3\left(t\right),x_4\left(\beta \left(t\right)\right),\dots ,x_n\left(\beta \left(t\right)\right)\right)}{{\partial }_{{\beta }_t}{x}_3\left(t\right)}{D}_{\beta }{x}_3\left(t\right)+\cdots +\frac{\partial f\left(x_1\left(t\right),x_2\left(t\right),\dots ,x_n\left(t\right)\right)}{{\partial }_{{\beta }_t}{x}_n\left(t\right)}{D}_{\beta }{x}_n\left(t\right).\hfill \end{array}$$

(11)

2.2. The $\beta$-Directional and the $\beta$-Gradient Derivatives

In this section, we introduce a generalization of the $\beta$-partial derivative which is the $\beta$-directional derivative by giving the rate of change of the function $f:I\times I\to \mathbb{R}$ at a point $P=(p_1,p_2)\in {\mathbb{R}}^2$ in the direction of the unit vector $\overrightarrow{v}=(v_1,v_2)$. Furthermore, we define the $\beta$-gradient vector of a multi-variable function, which can be regarded as another version of the $\beta$-directional derivative. In the following, for these two different approaches of the concept of the $\beta$-directional derivative, we will obtain $\beta$-analogues of some of the known properties of the classical directional derivative’s concept and also refer to some properties that remain accordingly to the considered definition.

Definition 2.

Let $f:I\times I\to \mathbb{R}$ be a function of two variables. Then, we define the β-directional derivative at the point $P=(p_1,p_2)$ in the direction of the unit vector $\overrightarrow{v}=(v_1,v_2)$, denoted by $D_{\beta ,\overrightarrow{v}}$, as

$$D_{\beta ,\overrightarrow{v}}f\left(P\right)\equiv D_{\beta ,\overrightarrow{v}}{f|}_P:=D_{\beta }f(P+t\overrightarrow{v}).$$

(12)

Theorem 1.

Let $f:I\times I\to \mathbb{R}$ be a function of two variables. Then, the β-directional derivative of f at a point $P=(p_1,p_2)$ in the direction of a unit vector $\overrightarrow{v}=(v_1,v_2)$ is given by

$$\begin{array}{ccc}{D}_{\beta ,\overrightarrow{v}}{f|}_P& =& {\displaystyle \frac{\partial f\left(x\left(t\right),y\left(\beta \right(t)\right)}{{\partial }_{{\beta }_t}x\left(t\right)}{v}_1+\frac{\partial f\left(x\left(t\right),y\left(t\right)\right)}{{\partial }_{{\beta }_t}y\left(t\right)}{v}_2}\\ & =& {\displaystyle \frac{\partial f\left(x\left(t\right),y\left(t\right)\right)}{{\partial }_{{\beta }_t}x\left(t\right)}{v}_1+\frac{\partial f\left(x\left(\beta \right(t\left)\right),y\left(t\right)\right)}{{\partial }_{{\beta }_t}y\left(t\right)}{v}_2,}\end{array}$$

(13)

where $x\left(t\right)=p_1+t{v}_1$ and $y\left(t\right)=p_2+t{v}_2$.

Proof. 

From (12), we have

$$D_{\beta ,\overrightarrow{v}}{f|}_P=D_{\beta }f(p_1+t{v}_1,p_2+t{v}_2).$$

Being $x\left(t\right)=p_1+t{v}_1$ and $y\left(t\right)=p_2+t{v}_2$ then $D_{\beta }x\left(t\right)=v_1$ and $D_{\beta }y\left(t\right)=v_2$. Hence, using Lemma 1, the result holds. □

Remark 3.

The proof of Theorem 1 can be performed directly by computing the right side of (13): in fact, from notations (7) and (8), we may write

$$\begin{array}{c}\frac{\partial f\left(x\left(t\right),y\left(\beta \right(t)\right)}{{\partial }_{{\beta }_t}x\left(t\right)}{v}_1+\frac{\partial f\left(x\left(t\right),y\left(t\right)\right)}{{\partial }_{{\beta }_t}y\left(t\right)}{v}_2=\hfill \\ \frac{f\left(x\left(\beta \right(t\left)\right),y\left(\beta \right(t)\right)-f\left(x\left(t\right),y\left(\beta \right(t)\right)}{x\left(\beta \right(t)-x(t)}{v}_1+\frac{f\left(x\left(t\right),y\left(\beta \right(t)\right)-f\left(x\left(t\right),y\left(t\right)\right)}{y\left(\beta \right(t)-y(t)}{v}_2.\hfill \end{array}$$

Since $x\left(\beta \left(t\right)\right)-x\left(t\right)=p_1+\beta \left(t\right)v_1-\left(p_1+t{v}_1\right)=\left(\beta \left(t\right)-t\right)v_1$ and $y\left(\beta \left(t\right)\right)-y\left(t\right)=p_2+\beta \left(t\right)v_2-\left(p_2+t{v}_2\right)=\left(\beta \left(t\right)-t\right)v_2$, we obtain

$$\frac{\partial f\left(x\left(t\right),y\left(\beta \right(t)\right)}{{\partial }_{{\beta }_t}x\left(t\right)}{v}_1+\frac{\partial f\left(x\left(t\right),y\left(t\right)\right)}{{\partial }_{{\beta }_t}y\left(t\right)}{v}_2=\frac{f\left(x\left(\beta \right(t\left)\right),y\left(\beta \right(t)\right)-f\left(x\left(t\right),y\left(t\right)\right)}{\beta \left(t\right)-t},$$

which, because $x\left(t\right)=p_1+t{v}_1$ and $y\left(t\right)=p_2+t{v}_2$, is exactly (12).

Theorem 2.

Let $f,g:I\times I\to \mathbb{R}$ be two functions and $\overrightarrow{v}=(v_1,v_2)\in {\mathbb{R}}^2$ a unit vector. Then, for any real numbers $\lambda$ and $\mu$,

$$D_{\beta ,\overrightarrow{v}}(\lambda f+{\mu g)|}_P=\lambda D_{\beta ,\overrightarrow{v}}f{{|}_P+\mu D_{\beta ,\overrightarrow{v}}g|}_P.$$

Proof. 

Using Equation (13), we have

$$D_{\beta ,\overrightarrow{v}}{(f+g)|}_P=\frac{\partial (f+g)\left(x\left(t\right),y\left(\beta \right(t)\right)}{{\partial }_{{\beta }_t}x\left(t\right)}{v}_1+\frac{\partial (f+g)\left(x\left(t\right),y\left(t\right)\right)}{{\partial }_{{\beta }_t}y\left(t\right)}{v}_2,$$

which, by (7) and (8), becomes

$$\begin{array}{c}{D}_{\beta ,\overrightarrow{v}}{(f+g)|}_P=\hfill \\ =\frac{\partial f\left(x\left(t\right),y\left(\beta \right(t)\right)}{{\partial }_{{\beta }_t}x\left(t\right)}{v}_1+\frac{\partial g\left(x\left(t\right),y\left(\beta \right(t)\right)}{{\partial }_{{\beta }_t}x\left(t\right)}{v}_1+\frac{\partial f\left(x\left(t\right),y\left(t\right)\right)}{{\partial }_{{\beta }_t}y\left(t\right)}{v}_2+\frac{\partial g\left(x\left(t\right),y\left(t\right)\right)}{{\partial }_{{\beta }_t}y\left(t\right)}{v}_2\hfill \\ =\frac{\partial f\left(x\left(t\right),y\left(\beta \right(t)\right)}{{\partial }_{{\beta }_t}x\left(t\right)}{v}_1+\frac{\partial f\left(x\left(t\right),y\left(t\right)\right)}{{\partial }_{{\beta }_t}y\left(t\right)}{v}_2+\frac{\partial g\left(x\left(t\right),y\left(\beta \right(t)\right)}{{\partial }_{{\beta }_t}x\left(t\right)}{v}_1+\frac{\partial g\left(x\left(t\right),y\left(t\right)\right)}{{\partial }_{{\beta }_t}y\left(t\right)}{v}_2.\hfill \end{array}$$

Thus, again by (13),

$$D_{\beta ,\overrightarrow{v}}{(f+g)|}_P=D_{\beta ,\overrightarrow{v}}f{{|}_P+D_{\beta ,\overrightarrow{v}}g|}_P.$$

Finally, using (13) together with (7) and (8), one proves that $D_{\beta ,\overrightarrow{v}}\left(\lambda f\right){{|}_P=\lambda D_{\beta ,\overrightarrow{v}}f|}_P.$ □

Again, we can state Theorem 1 and 2 in $n$ dimensions.

Theorem 3.

Let $f:I^n\to \mathbb{R}$ be a function of $n$ real variables $(x_1,\dots ,x_n),$ where each one of these variables $x_i:I\to I$ is regarded as a real function, for $i=1,\dots ,n$. Then, the β-directional derivative of $f$ at a point $P=(p_1,\dots ,p_n)\in {\mathbb{R}}^n$ in the direction of the unit vector $\overrightarrow{v}=(v_1,\dots ,v_n)\in {\mathbb{R}}^n$ is given by

$$\begin{array}{c}{D}_{\beta ,\overrightarrow{v}}{f|}_P=\frac{\partial f\left(x_1\left(t\right),x_2\left(\beta \left(t\right)\right),\dots ,x_n\left(\beta \left(t\right)\right)\right)}{{\partial }_{{\beta }_t}{x}_1\left(t\right)}{v}_1+\frac{\partial f\left(x_1\left(t\right),x_2\left(t\right),x_3\left(\beta \left(t\right)\right),\dots ,x_n\left(\beta \left(t\right)\right)\right)}{{\partial }_{{\beta }_t}{x}_2\left(t\right)}{v}_2\hfill \\ +\frac{\partial f\left(x_1\left(t\right),x_2\left(t\right),x_3\left(t\right),x_4\left(\beta \left(t\right)\right),\dots ,x_n\left(\beta \left(t\right)\right)\right)}{{\partial }_{{\beta }_t}{x}_3\left(t\right)}{v}_3+\dots +\frac{\partial f\left(x_1\left(t\right),x_2\left(t\right),\dots ,x_n\left(t\right)\right)}{{\partial }_{{\beta }_t}{x}_n\left(t\right)}{v}_n,\hfill \end{array}$$

where$x_i\left(t\right)=p_i+t{v}_i$for$i=1,2,\dots ,n$.

Proof. 

The proof is just a consequence of identity (11). □

Theorem 4.

Let $f,g:I^n\to \mathbb{R}$ be two functions of $n$ variables and $\overrightarrow{v}=(v_1,\dots ,v_n)\in {\mathbb{R}}^n$ be a unit vector. Then, for any real numbers $\lambda$ and $\mu$,

$$D_{\beta ,\overrightarrow{v}}(\lambda f+{\mu g)|}_P=\lambda D_{\beta ,\overrightarrow{v}}f{{|}_P+\mu D_{\beta ,\overrightarrow{v}}g|}_P.$$

Proof. 

The proof can be carried out in a similar way to the proof of Theorem 2. □

Now, we will define the $\beta$-gradient vector of a multi-variable function $f$.

Definition 3.

Let $f\left(x_1,x_2,\dots ,x_n\right)$ be a function of n-variables. Then, the β-gradient vector of the function $f$ at the point $P=(p_1,p_2,\dots ,p_n)$, denoted by ${grad}_{\beta }f\left(P\right)$, is defined by

$$\begin{array}{ccc}{grad}_{\beta }f\left(P\right)\hfill & \equiv \hfill & {▽}_{\beta }f\left(P\right)\equiv {▽}_{\beta }f(p_1,p_2,\dots ,p_n)\hfill \\ \hfill & =:\hfill & <\frac{\partial f(p_1,p_2,\dots ,p_n)}{{\partial }_{\beta }{x}_1},\frac{\partial f(p_1,p_2,\dots ,p_n)}{{\partial }_{\beta }{x}_2},\dots ,\frac{\partial f(p_1,p_2,\dots ,p_n)}{{\partial }_{\beta }{x}_n}>,\hfill \end{array}$$

where$p_1,p_2,\dots ,p_n\in I\backslash \left\{s_0\right\}$.

This definition enables us to consider another possible definition for the $\beta$-directional derivative.

Definition 4.

The β-gradient directional derivative of a multi-variable function $f$ at a point $P=(p_1,p_2,\dots ,p_n)$, in the direction of the unit vector $\overrightarrow{v}=(v_1,v_2,\dots ,v_n)$, is the dot product of the vectors ${▽}_{\beta }f(p_1,p_2,\dots ,p_n)$ and $\overrightarrow{v}$, which means that

$$\begin{array}{ccc}{\mathcal{D}}_{\beta ,\overrightarrow{v}}f\left(P\right)\hfill & :=\hfill & {▽}_{\beta }f\left(P\right)·\overrightarrow{v}\hfill \\ \hfill & :=\hfill & \frac{\partial f(p_1,p_2,\dots ,p_n)}{{\partial }_{\beta }{x}_1}{v}_1+\frac{\partial f(p_1,p_2,\dots ,p_n)}{{\partial }_{\beta }{x}_2}{v}_2+\cdots +\frac{\partial f(p_1,p_2,\dots ,p_n)}{{\partial }_{\beta }{x}_n}{v}_n.\hfill \end{array}$$

(14)

Remark 4.

• Of course, this latter definition keeps naturally and trivially some of the known properties of the classical directional derivative. For instance, the linearity property follows at once from the known properties of the dot product, and the β-analogue property of the classical computation of the directional derivative, when the function is differentiable at a given point, follows as a consequence of the definition itself and the observation that follows (4) and (5) or that follows (6).
• In the particular case where $\overrightarrow{v}=(0,0,\dots ,1_i,\dots ,0)$ is the unit vector, then the β-gradient directional derivative (14) gives the β-partial derivative with respect to the variable $x_i$.

2.3. Hahn-Directional Derivative

In this section, from the general quantum operator (2), we deduce the Hahn-partial and Hahn-directional derivatives, corresponding to the Hahn difference operator (3).

Let $f:I\times I\to \mathbb{R}$, $\beta \left(t\right)=qt+\omega$, $t\in I\subset \mathbb{R}$, $q\in (0,1)$, and $\omega >0$. Then, from (4) and (5), the Hahn-partial derivative of the function $f(x,y)$ in the direction of x and y, respectively, is given by

$$\frac{\partial f(x,y)}{{\partial }_{q,\omega }x}=\frac{f(qx+\omega ,y)-f(x,y)}{x(q-1)+\omega },$$

and

$$\frac{\partial f(x,y)}{{\partial }_{q,\omega }y}=\frac{f(x,qy+\omega )-f(x,y)}{y(q-1)+\omega },$$

where $x,y\in I$, $x\ne \frac{\omega }{1-q}$, and $y\ne \frac{\omega }{1-q}$.

The product rule and the quotient rule of the Hahn-partial derivative with respect to $x$, $x\ne s_0$, will have the following form:

$$\begin{array}{cc}\hfill \frac{\partial }{{\partial }_{q,\omega }x}\left(fg\right)(x,y)& =\left[\frac{\partial }{{\partial }_{q,\omega }x}f(x,y)\right]g(x,y)+f(qx+\omega ,y)\left[\frac{\partial }{{\partial }_{q,\omega }x}g(x,y)\right]\hfill \\ & =\left[\frac{\partial }{{\partial }_{q,\omega }x}f(x,y)\right]g\left(qx+\omega ,y\right)+f(x,y)\left[\frac{\partial }{{\partial }_{q,\omega }x}g(x,y)\right],\hfill \end{array}$$

where $\left(fg\right)(x,y)=f(x,y)g(x,y)$, and

$$\frac{\partial }{{\partial }_{q,\omega }x}\left(f/g\right)(x,y)=\frac{\left[\frac{\partial }{{\partial }_{q,\omega }x}f(x,y)\right]g(x,y)-f(x,y)\left[\frac{\partial }{{\partial }_{q,\omega }x}g(x,y)\right]}{g(x,y)g(qx+\omega ,y)},$$

where $g(x,y)g(qx+\omega ,y)\ne 0$, respectively. The cases corresponding to the Hahn-partial derivative with respect to the variable $y$, $y\ne s_0$, have a similar form.

Lemma 3.

Let $f:I\times I\to \mathbb{R}$ be a function of two variables and let $x,y:I\to I$ be two real functions of t. Then, $f(x,y)$ is a real function of $t$ and

$$D_{q,\omega }f(x\left(t\right),y\left(t\right))=\frac{\partial f\left(x\left(t\right),y(qt+\omega )\right)}{{\partial }_{q,\omega }x\left(t\right)}{D}_{q,\omega }x\left(t\right)+\frac{\partial f\left(x\left(t\right),y\left(t\right)\right)}{{\partial }_{q,\omega }y\left(t\right)}{D}_{q,\omega }y\left(t\right).$$

Furthermore, we can define the Hahn-directional derivative at a point $P=(p_1,p_2)$ in the direction of a unit vector $\overrightarrow{v}=(v_1,v_2)$ by

$$\frac{\partial f}{{\partial }_{q,\omega }\overrightarrow{v}}\left(P\right)\equiv \frac{\partial f}{{\partial }_{q,\omega }\overrightarrow{v}}{|}_P:=D_{q,\omega }f(P+t\overrightarrow{v}),$$

and, from Lemma 3, it is given by

$$\frac{\partial f}{{\partial }_{q,\omega }\overrightarrow{v}}{|}_P=\frac{\partial f\left(x\left(t\right),y\left(\beta \right(t)\right)}{{\partial }_{q,\omega }x\left(t\right)}{v}_1+\frac{\partial f\left(x\right(t),y(t\left)\right)}{{\partial }_{q,\omega }y\left(t\right)}{v}_2,$$

where $x\left(t\right)=p_1+t{v}_1$, $y\left(t\right)=p_2+t{v}_2$ and $\beta \left(t\right)=qt+\omega$.

Moreover, let $f,g:I\times I\to \mathbb{R}$ be two real functions, $\lambda ,\mu \in \mathbb{R}$, and let $\overrightarrow{v}=(v_1,v_2)$ be a unit vector. Then,

$$\frac{\partial \left(\lambda f+\mu g\right)\left(P\right)}{{\partial }_{q,\omega }\overrightarrow{v}}=\lambda \frac{\partial f\left(P\right)}{{\partial }_{q,\omega }\overrightarrow{v}}+\mu \frac{\partial g\left(P\right)}{{\partial }_{q,\omega }\overrightarrow{v}}$$

Definition 5.

Let $f:I\times I\to \mathbb{R}$ be a function of two variables. Then, the Hahn-gradient vector of the function f at $(x,y)$ is denoted by ${▽}_{q,\omega }f(x,y)$ and is defined by

$${▽}_{q,\omega }f(x,y):=<\frac{\partial f(x,y)}{{\partial }_{q,\omega }x},\frac{\partial f(x,y)}{{\partial }_{q,\omega }y}>.$$

Also, the Hahn-gradient directional derivative is given by

$${\mathcal{D}}_{(q,\omega ),\overrightarrow{v}}f(x,y):={▽}_{q,\omega }f(x,y)·\overrightarrow{v},$$

where$\overrightarrow{v}$is a unit vector in${\mathbb{R}}^2$.

It is worth mentioning that we can also obtain these results in the power quantum calculus by using $\beta \left(t\right)=q{t}^n$ where $n$ is an odd number.

Example 1.

Let $f(x,y)=x^2+2y$, and let $\beta \left(t\right)=\frac{1}{2}t+1$, $t\in I\subset \mathbb{R}$. Then,

$$\frac{\partial f(x,y)}{{\partial }_{q,\omega }x}=\frac{f(\frac{1}{2}x+1,y)-f(x,y)}{\frac{1}{2}x+1-x}=\frac{-\frac{3}{4}{x}^2+x+1}{-\frac{1}{2}x+1}.$$

and

$$\frac{\partial f(x,y)}{{\partial }_{q,\omega }y}=\frac{f(x,\frac{1}{2}y+1)-f(x,y)}{\frac{1}{2}y+1-y}=\frac{-y+2}{-\frac{1}{2}y+1}.$$

In addition,

$$▽f(x,y)=<\frac{\partial f(x,y)}{{\partial }_{q,\omega }x},\frac{\partial f(x,y)}{{\partial }_{q,\omega }y}>=<\frac{-\frac{3}{4}{x}^2+x+1}{-\frac{1}{2}x+1},\frac{-y+2}{-\frac{1}{2}y+1}>.$$

Example 2.

If we consider the function

$$g(x,y)=\left\{\begin{array}{ccc}0& if& (x,y)=(0,0)\\ \frac{x{y}^2}{x^2+y^2}& if& (x,y)\ne (0,0)\end{array}\right.,$$

then, for the classical theory, we have the following results:

• $g$ is a continuous function in ${\mathbb{R}}^2$;
• Whenever $(x,y)\ne (0,0)$,

  $$\frac{\partial g}{\partial x}(x,y)=\frac{y^2\left(y^2-x^2\right)}{{\left(x^2+y^2\right)}^2},\frac{\partial g}{\partial y}(x,y)=\frac{2x^{3}y}{{\left(x^2+y^2\right)}^2}$$

  and

  $$\frac{\partial g}{\partial x}(0,0)=0,\frac{\partial g}{\partial y}(0,0)=0;$$
• The directional derivative of $g$ in the direction of the vector $\overrightarrow{v}=(1,1)$ at the point $(0,0)$ is given by ${\displaystyle D_{\overrightarrow{v}}g(0,0)=\frac{1}{2};}$
• Joining these last two items, we may conclude that the function $g$ is not differentiable at the point $(0,0)$;
• Notice that, for instance, along the vertical line $x=0$ (with the exception of the origin $(0,0)$), we have

  $$\frac{\partial g}{\partial x}(0,y)=\frac{y^4}{y^4}=1and\frac{\partial g}{\partial y}(0,y)=0.$$

For the case of the general quantum operator $D_{\beta }$ (with a unique fixed point $s_0$), if we consider the corresponding function

$$h(x,y)=\left\{\begin{array}{ccc}0& if& (x,y)=(s_0,s_0)\\ \frac{(x-s_0){(y-s_0)}^2}{{(x-s_0)}^2+{(y-s_0)}^2}& if& (x,y)\ne (s_0,s_0)\end{array}\right.$$

then, whenever$(x,y)\ne (s_0,s_0)$, we have

$$\frac{\partial h}{{\partial }_{\beta }x}(x,y)=\frac{\left({(y-s_0)}^2-(x-s_0)(\beta \left(x\right)-s_0)\right){(y-s_0)}^2}{\left({\left(\beta \left(x\right)-s_0\right)}^2+{(y-s_0)}^2\right)\left({(x-s_0)}^2+{(y-s_0)}^2\right)}$$

and

$$\frac{\partial h}{{\partial }_{\beta }y}(x,y)=\frac{\left((y-s_0)+(\beta \left(y\right)-s_0)\right){(x-s_0)}^3}{\left({(x-s_0)}^2+{\left(\beta \left(y\right)-s_0\right)}^2\right)\left({(x-s_0)}^2+{(y-s_0)}^2\right)}.$$

Notice that, for instance, along the vertical line $x=s_0$ (with the exception of the point $(s_0,s_0)$), we have

$$\frac{\partial h}{{\partial }_{\beta }x}(s_0,y)=\frac{{(y-s_0)}^4}{{(y-s_0)}^4}=1and\frac{\partial h}{{\partial }_{\beta }y}(s_0,y)=0.$$

We note that for the case of the function $g$, we have

$$\frac{\partial g}{{\partial }_{\beta }x}(x,y)=\frac{\left(y^2-x\beta \left(x\right)\right)y^2}{\left({\left(\beta \left(x\right)\right)}^2+y^2\right)\left(x^2+y^2\right)},\frac{\partial g}{{\partial }_{\beta }y}(x,y)=\frac{\left(y+\beta \left(y\right)\right)x^3}{\left(x^2+{\left(\beta \left(y\right)\right)}^2\right)\left(x^2+y^2\right)}.$$

For the Jackson’s quantum operator, where $\beta \left(t\right)=qt$ with $q$ fixed in the interval $(0,1)$, its fixed point is $s_0=0$.

• From the above observations, we have ${\displaystyle \frac{\partial g}{{\partial }_{\beta }x}(0,0)=\frac{\partial g}{\partial x}(0,0)=0}$ and ${\displaystyle \frac{\partial g}{{\partial }_{\beta }y}(0,0)=\frac{\partial g}{\partial y}(0,0)=0.}$
• Whenever $(x,y)\ne (0,0)$, we have ${\displaystyle \frac{\partial g}{{\partial }_{\beta }x}(x,y)=\frac{\left(y^2-q{x}^2\right)y^2}{\left(q^2{x}^2+y^2\right)\left(x^2+y^2\right)}}$ and ${\displaystyle \frac{\partial g}{{\partial }_{\beta }y}(x,y)=\frac{(1+q)x^{3}y}{\left(x^2+q^2{y}^2\right)\left(x^2+y^2\right)}.}$

3. Conclusions

In this paper, we introduced the $\beta$-partial derivative and the $\beta$-directional derivative of a multi-variable function associated with the $\beta$-difference operator, $D_{\beta }$, which is defined by ${\displaystyle D_{\beta }f\left(t\right)=\left(f\left(\beta \right(t\left)\right)-f\left(t\right)\right)/\left(\beta \left(t\right)-t\right)}$. We defined the $\beta$-gradient vector and the $\beta$-gradient directional derivative of a multi-variable function as well. We also deduced the Hahn-partial and the Hahn-directional derivatives associated with the Hahn difference operator. Finally, we proved some properties with some applications by giving some examples.