Editing Difference quotient (section)

==Defining the point range==
Regardless if ΔP is infinitesimal or finite, there is (at least—in the case of the derivative—theoretically) a point range, where the boundaries are P&nbsp;±&nbsp;(0.5)&nbsp;ΔP (depending on the orientation—ΔF(P), δF(P) or ∇F(P)):
:LB = Lower Boundary;  UB = Upper Boundary;
Derivatives can be regarded as functions themselves, harboring their own derivatives. Thus each function is home to sequential degrees ("higher orders") of derivation, or ''differentiation''.  This property can be generalized to all difference quotients.<br>
As this sequencing requires a corresponding boundary splintering, it is practical to break up the point range into smaller, equi-sized sections, with each section being marked by an intermediary point (''P''<sub>''i''</sub>), where LB = ''P''<sub>0</sub> and UB = ''P''<sub>''ń''</sub>, the ''n''th  point, equaling the degree/order:
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   LB =  P<sub>0</sub>  = P<sub>0</sub> + 0Δ<sub>1</sub>P     = P<sub>ń</sub> − (Ń-0)Δ<sub>1</sub>P;
         P<sub>1</sub>  = P<sub>0</sub> + 1Δ<sub>1</sub>P     = P<sub>ń</sub> − (Ń-1)Δ<sub>1</sub>P;
         P<sub>2</sub>  = P<sub>0</sub> + 2Δ<sub>1</sub>P     = P<sub>ń</sub> − (Ń-2)Δ<sub>1</sub>P;
         P<sub>3</sub>  = P<sub>0</sub> + 3Δ<sub>1</sub>P     = P<sub>ń</sub> − (Ń-3)Δ<sub>1</sub>P;
             ↓      ↓        ↓       ↓
        P<sub>ń-3</sub> = P<sub>0</sub> + (Ń-3)Δ<sub>1</sub>P = P<sub>ń</sub> − 3Δ<sub>1</sub>P;
        P<sub>ń-2</sub> = P<sub>0</sub> + (Ń-2)Δ<sub>1</sub>P = P<sub>ń</sub> − 2Δ<sub>1</sub>P;
        P<sub>ń-1</sub> = P<sub>0</sub> + (Ń-1)Δ<sub>1</sub>P = P<sub>ń</sub> − 1Δ<sub>1</sub>P;
   UB = P<sub>ń-0</sub> = P<sub>0</sub> + (Ń-0)Δ<sub>1</sub>P = P<sub>ń</sub> − 0Δ<sub>1</sub>P = P<sub>ń</sub>;

   ΔP = Δ<sub>1</sub>P = P<sub>1</sub> − P<sub>0</sub> = P<sub>2</sub> − P<sub>1</sub> = P<sub>3</sub> − P<sub>2</sub> = ... = P<sub>ń</sub> − P<sub>ń-1</sub>;

   ΔB = UB − LB = P<sub>ń</sub> − P<sub>0</sub> = Δ<sub>ń</sub>P = ŃΔ<sub>1</sub>P.