## 3 Statistical Deferred Cesaro Derivative  
The derivative of a function \(\eta:\mathbb{R}\to\mathbb{R}\) at a point \(v_{0}\) can be defined using two fundamental formulas, both of which involve a positive sequence \((v_{\rho})\) such that \(\lim_{\rho\to\infty}v_{\rho}=0\). These formulas are given as follows:  
\[\eta^{\prime}(v_{0})=\lim_{\rho\to\infty}\frac{\eta(v_{0}+v_{\rho})-\eta(v_{0} )}{v_{\rho}}\ \ \text{and}\ \ \eta^{\prime}(v_{0})=\lim_{\rho\to\infty}\frac{\eta(v_{0}+v_{\rho})-\eta(v_{0} -v_{\rho})}{2v_{\rho}}.\]  
The initial formula, referred to as [PERSON]'s difference quotient, represents the slope of a secant line to the graph of \(\eta\). Meanwhile, the second formula, known as the symmetric difference quotient, calculates the slope of a chord on the graph of \(\eta\). For additional information, see [24]. Using a similar methodology, we now extend the definition to encompass the derivative of the deferred Cesaro summability mean. **Definition 3**. : _A function \(\eta:\mathbb{R}\to\mathbb{R}\) is said to possess a deferred Cesaro derivative \(\kappa\in\mathbb{R}\) at a point \(v_{0}\in\mathbb{R}\) if_  
\[\lim_{\rho\to\infty}\frac{1}{b_{\rho}-a_{\rho}}\sum_{i=a_{\rho}+1}^{b_{\rho}} \frac{\eta(v_{0}+v_{i})-\eta(v_{0})}{v_{i}}=\kappa,\]  
_where \(v_{0}>0\) and \(\lim_{\rho\to\infty}v_{\rho}=0\)._  
An equivalent formulation of Definition 3 is presented as follows:  
**Definition 4**. : _A function \(\eta:\mathbb{R}\to\mathbb{R}\) is said to possess a deferred Cesaro derivative \(\kappa\in\mathbb{R}\) at a point \(v_{0}\in\mathbb{R}\) if_  
\[\lim_{\rho\to\infty}\frac{1}{b_{\rho}-a_{\rho}}\sum_{i=a_{\rho}+1}^{b_{\rho}} \frac{\eta(v_{0}+v_{i})-\eta(v_{0}-v_{i})}{2v_{i}}=\kappa\]  
_holds, whenever \(v_{0}>0\) and \(\lim_{\rho\to\infty}v_{\rho}=0\)._  
Next, we introduce the definition and an illustrative example of the statistical derivative for the deferred Cesaro mean as follows:  
**Definition 5**. : _A function \(\eta:\mathbb{R}\to\mathbb{R}\) is said to possess a statistical deferred Cesaro derivative \(\kappa\in\mathbb{R}\) at a point \(v_{0}\in\mathbb{R}\) if_  
\[\lim_{\rho\to\infty}\frac{1}{b_{\rho}-a_{\rho}}\bigg{|}\bigg{\{}a_{\rho} \leqq b_{\rho}\ \ \text{and}\ \ \bigg{|}\frac{\eta(v_{0}+v_{\rho})-\eta(v_{0})}{v_{\rho}}-\kappa\bigg{|} \geqq\epsilon\bigg{\}}\bigg{|}=0\]  
_holds, whenever \(v_{0}>0\) and \(\lim_{\rho\to\infty}v_{\rho}=0\)._  
_In this context, we express it as follows:_  
\[\mathrm{DCD}_{\mathrm{stat}}\eta(v_{\rho})=\kappa.\]  
**Example 1**. : _Consider the function \(\eta(v)=v^{2}\), and let \(v_{0}=1\). Define the sequences \((a_{\rho})=\rho\), \((b_{\rho})=2\rho\), and \((v_{\rho})=\frac{1}{\rho}\). We aim to verify whether \(\eta\) has a statistical deferred Cesaro derivative \(\kappa\) at \(v_{0}=1\). For \(\eta(v)=v^{2}\), the difference quotient is given by_  
\[\frac{\eta(v_{0}+v_{\rho})-\eta(v_{0})}{v_{\rho}}=\frac{\left(1+\frac{1}{\rho }\right)^{2}-1^{2}}{\frac{1}{\rho}}=2+\frac{1}{\rho}.\]  
_As \(\rho\to\infty\), the difference quotient converges to \(\kappa=2\). In view of the statistical deferred Cesaro derivative,_  
\[\lim_{\rho\to\infty}\frac{1}{b_{\rho}-a_{\rho}}\left|\left\{a_{\rho}\leq b_{ \rho}\ \ \text{and}\ \ \left|\frac{\eta(v_{0}+v_{\rho})-\eta(v_{0})}{v_{\rho}}-\kappa\right|\geqq \epsilon\right\}\right|=0,\]  
_for a given \(\epsilon>0\), we have_  
\[\left|\frac{\eta(v_{0}+v_{\rho})-\eta(v_{0})}{v_{\rho}}-\kappa\right|=\frac{1 }{\rho},\]  
_which becomes arbitrarily small as \(\rho\to\infty\). Hence, the set of \(\rho\) for which the inequality_  
\[\left|\frac{\eta(v_{0}+v_{\rho})-\eta(v_{0})}{v_{\rho}}-\kappa\right|\geqq \epsilon,\]  
_holds becomes negligible in the deferred Cesaro sense as \(\rho\to\infty\). Thus, \(\eta(v)=v^{2}\) has a statistical deferred Cesaro derivative \(\kappa=2\) at \(v_{0}=1\)._  
Figure 1 illustrates how the difference quotient, calculated for the sequence \((v_{\rho})=\frac{1}{\rho}\), approaches the true derivative value \(\kappa=2\) as \(\rho\to\infty\).