Editing Difference quotient (section)

==The primary difference quotient (''Ń'' = 1)==
:<math>\frac{\Delta F(P_0)}{\Delta P}=\frac{F(P_{\acute{n}})-F(P_0)}{\Delta_{\acute{n}}P}=\frac{F(P_1)-F(P_0)}{\Delta _1P}=\frac{F(P_1)-F(P_0)}{P_1-P_0}.\,\!</math>

===As a derivative===
:The difference quotient as a derivative needs no explanation, other than to point out that, since P<sub>0</sub> essentially equals P<sub>1</sub> = P<sub>2</sub> = ... = P<sub>ń</sub> (as the differences are infinitesimal), the [[Leibniz notation]] and derivative expressions do not distinguish P to P<sub>0</sub> or P<sub>ń</sub>:

:::<math>\frac{dF(P)}{dP}=\frac{F(P_1)-F(P_0)}{dP}=F'(P)=G(P).\,\!</math>
There are [[Derivative#Notation for differentiation|other derivative notations]], but these are the most recognized, standard designations.

===As a divided difference===
:A divided difference, however, does require further elucidation, as it equals the average derivative between and including LB and UB:

:: <math>
\begin{align}
P_{(tn)} & =LB+\frac{TN-1}{UT-1}\Delta B \ =UB-\frac{UT-TN}{UT-1}\Delta B; \\[10pt]
& {} \qquad {\color{white}.}(P_{(1)}=LB,\  P_{(ut)}=UB){\color{white}.} \\[10pt]
F'(P_\tilde{a}) & =F'(LB < P < UB)=\sum_{TN=1}^{UT=\infty}\frac{F'(P_{(tn)})}{UT}.
\end{align}
</math>

:In this interpretation, P<sub>ã</sub> represents a function extracted, average value of P (midrange, but usually not exactly midpoint), the particular valuation depending on the function averaging it is extracted from.  More formally, P<sub>ã</sub> is found in the [[mean value theorem]] of calculus, which says:

::''For any function that is continuous on [LB,UB] and differentiable on (LB,UB) there exists some P<sub>ã</sub> in the interval (LB,UB) such that the secant joining the endpoints of the interval [LB,UB] is parallel to the tangent at P<sub>ã</sub>.''

:Essentially, P<sub>ã</sub> denotes some value of P between LB and UB—hence,

::<math>P_\tilde{a}:=LB < P < UB=P_0 < P < P_\acute{n} \,\!</math>

:which links the mean value result with the divided difference:

:: <math>
\begin{align}
\frac{DF(P_0)}{DP} & = F[P_0,P_1]=\frac{F(P_1)-F(P_0)}{P_1-P_0}=F'(P_0 < P < P_1)=\sum_{TN=1}^{UT=\infty}\frac{F'(P_{(tn)})}{UT}, \\[8pt]
& = \frac{DF(LB)}{DB}=\frac{\Delta F(LB)}{\Delta B}=\frac{\nabla F(UB)}{\Delta B}, \\[8pt]
& = F[LB,UB]=\frac{F(UB)-F(LB)}{UB-LB}, \\[8pt]
& =F'(LB < P < UB)=G(LB < P < UB).
\end{align}
</math>

:As there is, by its very definition, a tangible difference between LB/P<sub>0</sub> and UB/P<sub>ń</sub>, the Leibniz and derivative expressions ''do'' require [[divaricate|divarication]] of the function argument.