# Unifying Aspects of Generalized Calculus
## 4 [PERSON] \(\kappa\)-Calculus Versus Non-Newtonian Calculus  
[PERSON], in a series of papers [26; 27; 28; 29; 30; 31; 32; 33; 34], developed a generalized form of arithmetic and calculus, with numerous applications to statistical physics, and beyond. In the present section, we will clarify links between his formalism and non-Newtonian calculus. As we will see, some of the results have a straightforward non-Newtonian interpretation, but not all. Assume \(\mathbb{X}=\mathbb{R}\), with the bijection \(f_{\mathbb{X}}\equiv f_{\kappa}:\mathbb{R}\to\mathbb{R}\) given explicitly by  
\[f_{\kappa}(x) = \frac{1}{\kappa}\arcsinh\,\kappa x, \tag{47}\] \[f_{\kappa}^{-1}(x) = \frac{1}{\kappa}\sinh\kappa x. \tag{48}\]  
[PERSON] \(\kappa\)-calculus begins with the arithmetic,  
\[x\stackrel{{\kappa}}{{\oplus}}y = f_{\kappa}^{-1}\big{(}f_{\kappa}(x)+f_{\kappa}(y)\big{)}, \tag{49}\] \[x\stackrel{{\kappa}}{{\ominus}}y = f_{\kappa}^{-1}\big{(}f_{\kappa}(x)-f_{\kappa}(y)\big{)},\] (50) \[x\stackrel{{\kappa}}{{\ominus}}y = f_{\kappa}^{-1}\big{(}f_{\kappa}(x)\cdot f_{\kappa}(y)\big{)},\] (51) \[x\stackrel{{\kappa}}{{\ominus}}y = f_{\kappa}^{-1}\big{(}f_{\kappa}(x)/f_{\kappa}(y)\big{)}. \tag{52}\]  
As \(f_{0}(x)=x\), the case \(\kappa=0\) corresponds to the usual field \(\mathbb{R}_{0}=(\mathbb{R},+,\cdot)\), which we will shortly denote by \(\mathbb{R}\). The neutral element of addition, \(0_{\kappa}=f_{\kappa}^{-1}(0)=0\), is the same for all \(\kappa\)s. The neutral element of \(\kappa\)-multiplication is nontrivial, \(1_{\kappa}=f_{\kappa}^{-1}(1)\
eq 1\). The fields \(\mathbb{R}_{\kappa}=(\mathbb{R},\stackrel{{\kappa}}{{\ominus}},\stackrel{{\kappa}}{{\ominus}})\) are isomorphic to one another due to their isomorphism with \(\mathbb{R}_{0}\),  
\[f_{\kappa}\big{(}x\stackrel{{\kappa}}{{\oplus}}y \big{)} = f_{\kappa}(x)+f_{\kappa}(y), \tag{53}\] \[f_{\kappa}\big{(}x\stackrel{{\kappa}}{{\ominus}}y \big{)} = f_{\kappa}(x)\cdot f_{\kappa}(y). \tag{54}\]  
[PERSON] defines his \(\kappa\)-derivative of a real function \(A(x)\) as  
\[\frac{\mathrm{d}A(x)}{\mathrm{d}_{\kappa}x} = \lim_{\delta\to 0}\frac{A(x+\delta)-A(x)}{(x+\delta) \stackrel{{\kappa}}{{\ominus}}x}=\frac{\mathrm{d}A(x)}{\mathrm{d} x}\Big{/}\frac{\mathrm{d}f_{\kappa}(x)}{\mathrm{d}x}=\frac{\mathrm{d}A(x)}{ \mathrm{d}x}\sqrt{1+\kappa^{2}x^{2}}. \tag{55}\]  
We will now specify in which sense the \(\kappa\)-derivative is non-Newtonian. First consider a function \(A\)\[ \begin{array}{ccc}\mathbb{R}_{\kappa_{1}}&\xrightarrow{A}&\mathbb{R}_{\kappa_{2}} \\ f_{\kappa_{1}}\Big{\downarrow}&&\Big{\downarrow}f_{\kappa_{2}}\\ \mathbb{R}&\xrightarrow{A}&\mathbb{R}\end{array}  \tag{56}\]  
Its non-Newtonian derivative  
\[\frac{\mathrm{D}A(x)}{\mathrm{D}x}=\lim_{\delta\to 0}\Big{(}A(x\oplus \delta_{\kappa_{1}})\stackrel{{\kappa_{2}}}{{\ominus}}A(x)\Big{)} \stackrel{{\kappa_{2}}}{{\ominus}}\delta_{\kappa_{2}}, \tag{57}\]  
if compared with (55), suggests \(\kappa_{2}=0\). Setting \(\kappa_{1}=\kappa\), \(\kappa_{2}=0\), we find  
\[\frac{\mathrm{D}A(x)}{\mathrm{D}x}=\lim_{\delta\to 0}\frac{A(x\stackrel{{ \kappa}}{{\oplus}}\delta_{\kappa})-A(x)}{\delta}=\lim_{\delta\to 0}\frac{A[x \stackrel{{\kappa}}{{\oplus}}f_{\kappa}^{-1}(\delta)]-A(x)}{ \delta}=\lim_{\delta\to 0}\frac{A(x\stackrel{{\kappa}}{{\oplus}} \delta)-A(x)}{\delta}, \tag{58}\]  
as \(f_{\kappa}^{-1}(\delta)\approx\delta\) for \(\delta\approx 0\). Denoting \(x\stackrel{{\kappa}}{{\oplus}}\delta=x+\delta^{\prime}\) we find \(\delta=(x+\delta^{\prime})\stackrel{{\kappa}}{{\ominus}}x\), and  
\[\frac{\mathrm{D}A(x)}{\mathrm{D}x}=\lim_{\delta^{\prime}\to 0}\frac{A(x+\delta^{ \prime})-A(x)}{(x+\delta^{\prime})\stackrel{{\kappa}}{{\ominus}}x}, \tag{59}\]  
in agreement with the Kaniadakis formula. However, as a by-product of the calculation we have proved that \(\kappa\)-calculus is applicable only to functions mapping \(\mathbb{R}_{\kappa}\) into \(\mathbb{R}\).