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| {{cleanup|reason=This article neglects the basic Wikipedia conventions of [[WP:MOS]] and [[WP:MOSMATH]].|date=August 2013}}
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| The primary vehicle of [[calculus]] and other higher mathematics is the [[Function (mathematics)|function]]. Its "input value" is its ''argument'', usually a point ("P") expressible on a graph. The difference between two points, themselves, is known as their ''[[Delta (letter)|Delta]]'' (Δ''P''), as is the difference in their function result, the particular notation being determined by the direction of formation:
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| *Forward difference: Δ''F''(''P'') = ''F''(''P'' + Δ''P'') − ''F''(''P'');
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| *Central difference: δF(P) = F(P + ½ΔP) − F(P − ½ΔP);
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| *Backward difference: ∇F(P) = F(P) − F(P − ΔP).
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| The general preference is the forward orientation, as F(P) is the base, to which differences (i.e., "ΔP"s) are added to it. Furthermore,
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|
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| *If |ΔP| is ''finite'' (meaning measurable), then ΔF(P) is known as a '''finite difference''', with specific denotations of DP and DF(P);
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| *If |ΔP| is ''[[infinitesimal]]'' (an infinitely small amount—''<math>\iota</math>''—usually expressed in standard analysis as a limit: <math>\lim_{\Delta P\rightarrow 0}\,\!</math>), then ΔF(P) is known as an '''infinitesimal difference''', with specific denotations of dP and dF(P) (in calculus graphing, the point is almost exclusively identified as "x" and F(x) as "y").
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| The function difference divided by the point difference is known as the '''''difference quotient''''' (attributed to [[Isaac Newton]],{{Citation needed|date=June 2007}} it is also known as ''Newton's quotient''):
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| :<math>\frac{\Delta F(P)}{\Delta P}=\frac{F(P+\Delta P)-F(P)}{\Delta P}=\frac{\nabla F(P+\Delta P)}{\Delta P}.\,\!</math>
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| If ΔP is infinitesimal, then the difference quotient is a ''[[derivative]]'', otherwise it is a ''[[divided differences|divided difference]]'':
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| :<math> \text{If } |\Delta P| = \mathit{ \iota}: \quad \frac{\Delta F(P)}{\Delta P}=\frac{dF(P)}{dP}=F'(P)=G(P);\,\!</math>
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| :<math> \text{If } |\Delta P| > \mathit{ \iota}: \quad \frac{\Delta F(P)}{\Delta P}=\frac{DF(P)}{DP}=F[P,P+\Delta P].\,\!</math>
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| ==Defining the point range==
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| 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 ± (0.5) ΔP (depending on the orientation—ΔF(P), δF(P) or ∇F(P)):
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| :LB = Lower Boundary; UB = Upper Boundary;
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| 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>
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| 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;
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| P<sub>1</sub> = P<sub>0</sub> + 1Δ<sub>1</sub>P = P<sub>ń</sub> − (Ń-1)Δ<sub>1</sub>P;
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| P<sub>2</sub> = P<sub>0</sub> + 2Δ<sub>1</sub>P = P<sub>ń</sub> − (Ń-2)Δ<sub>1</sub>P;
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| P<sub>3</sub> = P<sub>0</sub> + 3Δ<sub>1</sub>P = P<sub>ń</sub> − (Ń-3)Δ<sub>1</sub>P;
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| ↓ ↓ ↓ ↓
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| 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;
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| 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;
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| 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;
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| 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>;
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| 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>;
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| ΔB = UB − LB = P<sub>ń</sub> − P<sub>0</sub> = Δ<sub>ń</sub>P = ŃΔ<sub>1</sub>P.
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| ==The primary difference quotient (''Ń'' = 1)==
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| :<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>
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| ===As a derivative===
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| :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>:
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| :::<math>\frac{dF(P)}{dP}=\frac{F(P_1)-F(P_0)}{dP}=F'(P)=G(P).\,\!</math>
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| There are [[Derivative#Notations for differentiation|other derivative notations]], but these are the most recognized, standard designations.
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| ===As a divided difference===
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| :A divided difference, however, does require further elucidation, as it equals the average derivative between and including LB and UB:
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| :: <math>
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| \begin{align}
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| P_{(tn)} & =LB+\frac{TN-1}{UT-1}\Delta B \ =UB-\frac{UT-TN}{UT-1}\Delta B; \\[10pt]
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| & {} \qquad {\color{white}.}(P_{(1)}=LB,\ P_{(ut)}=UB){\color{white}.} \\[10pt]
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| F'(P_\tilde{a}) & =F'(LB < P < UB)=\sum_{TN=1}^{UT=\infty}\frac{F'(P_{(tn)})}{UT}.
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| \end{align}
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| </math>
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| | |
| :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:
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| ::''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>.''
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| :Essentially, P<sub>ã</sub> denotes some value of P between LB and UB—hence, | |
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| ::<math>P_\tilde{a}:=LB < P < UB=P_0 < P < P_\acute{n} \,\!</math>
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| :which links the mean value result with the divided difference:
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| :: <math>
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| \begin{align}
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| \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]
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| & = \frac{DF(LB)}{DB}=\frac{\Delta F(LB)}{\Delta B}=\frac{\nabla F(UB)}{\Delta B}, \\[8pt]
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| & = F[LB,UB]=\frac{F(UB)-F(LB)}{UB-LB}, \\[8pt]
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| & =F'(LB < P < UB)=G(LB < P < UB).
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| \end{align}
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| </math>
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| :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 divarication of the function argument.
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| ==Higher-order difference quotients==
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| ===Second order===
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| : <math>
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| \begin{align}
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| \frac{\Delta^2F(P_0)}{\Delta_1P^2} & =\frac{\Delta F'(P_0)}{\Delta_1P}=\frac{\frac{\Delta F(P_1)}{\Delta_1P}-\frac{\Delta F(P_0)}{\Delta_1P}}{\Delta_1P}, \\[10pt]
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| & =\frac{\frac{F(P_2)-F(P_1)}{\Delta_1P}-\frac{F(P_1)-F(P_0)}{\Delta_1P}}{\Delta_1P}, \\[10pt]
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| & =\frac{F(P_2)-2F(P_1)+F(P_0)}{\Delta_1P^2};
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| \end{align}
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| </math>
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| : <math>
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| \begin{align}
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| \frac{d^2F(P)}{dP^2} & = \frac{dF'(P)}{dP}=\frac{F'(P_1)-F'(P_0)}{dP}, \\[10pt]
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| & =\ \frac{dG(P)}{dP}=\frac{G(P_1)-G(P_0)}{dP}, \\[10pt]
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| & =\frac{F(P_2)-2F(P_1)+F(P_0)}{dP^2}, \\[10pt]
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| & =F''(P)=G'(P)=H(P)
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| \end{align}
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| </math>
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| : <math>
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| \begin{align}
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| \frac{D^2F(P_0)}{DP^2} & =\frac{DF'(P_0)}{DP}=\frac{F'(P_1 < P < P_2)-F'(P_0 < P < P_1)}{P_1-P_0}, \\[10pt]
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| & {\color{white}.} \qquad \ne\frac{F'(P_1)-F'(P_0)}{P_1-P_0}, \\[10pt]
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| & =F[P_0,P_1,P_2]=\frac{F(P_2)-2F(P_1)+F(P_0)}{(P_1-P_0)^2}, \\[10pt]
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| & =F''(P_0 < P < P_2)=\sum_{TN=1}^\infty \frac{F''(P_{(tn)})}{UT}, \\[10pt]
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| & =G'(P_0 < P < P_2)=H(P_0 < P < P_2).
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| \end{align}
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| </math>
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| | |
| ===Third order===
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| : <math>
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| \begin{align}
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| \frac{\Delta^3F(P_0)}{\Delta_1P^3} & = \frac{\Delta^2 F'(P_0)}{\Delta_1P^2}=\frac{\Delta F''(P_0)}{\Delta_1P}
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| =\frac{\frac{\Delta F'(P_1)}{\Delta_1P}-\frac{\Delta F'(P_0)}{\Delta_1P}}{\Delta_1P}, \\[10pt]
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| & =\frac{\frac{\frac{\Delta F(P_2)}{\Delta_1P}-\frac{\Delta F'(P_1)}{\Delta_1P}}{\Delta_1P}-
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| \frac{\frac{\Delta F'(P_1)}{\Delta_1P}-\frac{\Delta F'(P_0)}{\Delta_1P}}{\Delta_1P}}{\Delta_1P}, \\[10pt]
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| & =\frac{\frac{F(P_3)-2F(P_2)+F(P_1)}{\Delta_1P^2}-\frac{F(P_2)-2F(P_1)+F(P_0)}{\Delta_1P^2}}{\Delta_1P}, \\[10pt]
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| & =\frac{F(P_3)-3F(P_2)+3F(P_1)-F(P_0)}{\Delta_1P^3}; | |
| \end{align}
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| </math>
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| : <math>
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| \begin{align}
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| \frac{d^3F(P)}{dP^3} & =\frac{d^2F'(P)}{dP^2}=\frac{dF''(P)}{dP}=\frac{F''(P_1)-F''(P_0)}{dP}, \\[10pt]
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| & =\frac{d^2G(P)}{dP^2}\ =\frac{dG'(P)}{dP}\ =\frac{G'(P_1)-G'(P_0)}{dP}, \\[10pt]
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| & {\color{white}.}\qquad\qquad\ \ =\frac{dH(P)}{dP}\ =\frac{H(P_1)-H(P_0)}{dP}, \\[10pt]
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| & =\frac{G(P_2)-2G(P_1)+G(P_0)}{dP^2}, \\[10pt]
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| & =\frac{F(P_3)-3F(P_2)+3F(P_1)-F(P_0)}{dP^3}, \\[10pt]
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| & =F'''(P)=G''(P)=H'(P)=I(P);
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| \end{align}
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| </math>
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| : <math>
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| \begin{align}
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| \frac{D^3F(P_0)}{DP^3} & =\frac{D^2F'(P_0)}{DP^2}=\frac{DF''(P_0)}{DP}=\frac{F''(P_1 < P < P_3)-F''(P_0 < P < P_2)}{P_1-P_0}, \\[10pt]
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| & {\color{white}.}\qquad\qquad\qquad\qquad\qquad\ \ \ne\frac{F''(P_1)-F''(P_0)}{P_1-P_0}, \\[10pt]
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| & =\frac{\frac{F'(P_2 < P < P_3)-F'(P_1 < P < P_2)}{P_1-P_0}-\frac{F'(P_1 < P < P_2)-F'(P_0 < P < P_1)}{P_1-P_0}}{P_1-P_0}, \\[10pt]
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| & =\frac{F'(P_2 < P < P_3)-2F'(P_1 < P < P_2)+F'(P_0 < P < P_1)}{(P_1-P_0)^2}, \\[10pt]
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| & =F[P_0,P_1,P_2,P_3]=\frac{F(P_3)-3F(P_2)+3F(P_1)-F(P_0)}{(P_1-P_0)^3}, \\[10pt]
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| & =F'''(P_0 < P < P_3)=\sum_{TN=1}^{UT=\infty}\frac{F'''(P_{(tn)})}{UT}, \\[10pt]
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| & =G''(P_0 < P < P_3)\ =H'(P_0 < P < P_3)=I(P_0 < P < P_3).
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| \end{align}
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| </math>
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| | |
| ===''Ń''th order===
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| : <math>
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| \begin{align}
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| \Delta^\acute{n}F(P_0) & =F^{(\acute{n}-1)}(P_1)-F^{(\acute{n}-1)}(P_0), \\[10pt]
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| & =\frac{F^{(\acute{n}-2)}(P_2)-F^{(\acute{n}-2)}(P_1)}{\Delta_1P}-\frac{F^{(\acute{n}-2)}(P_1)-F^{(\acute{n}-2)}(P_0)}{\Delta_1P}, \\[10pt]
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| & =\frac{\frac{F^{(\acute{n}-3)}(P_3)-F^{(\acute{n}-3)}(P_2)}{\Delta_1P}-\frac{F^{(\acute{n}-3)}(P_2)-F^{(\acute{n}-3)}(P_1)}{\Delta_1P}}{\Delta_1P} \\[10pt]
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| & {\color{white}.}\qquad -\frac{\frac{F^{(\acute{n}-3)}(P_2)-F^{(\acute{n}-3)}(P_1)}{\Delta_1P}-\frac{F^{(\acute{n}-3)}(P_1)-F^{(\acute{n}-3)}(P_0)}{\Delta_1P}}{\Delta_1P}, \\[10pt]
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| & = \cdots
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| \end{align}
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| </math>
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| : <math>
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| \begin{align}
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| \frac{\Delta^\acute{n}F(P_0)}{\Delta_1P^\acute{n}} & =\frac{\sum_{I=0}^{\acute{N}}{-1\choose\acute{N}-I}{\acute{N}\choose I}F(P_0+I\Delta_1P)}{\Delta_1P^\acute{n}}; \\[10pt]
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| & \frac{\nabla^\acute{n}F(P_\acute{n})}{\Delta_1P^\acute{n}} \\[10pt]
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| & =\frac{\sum_{I=0}^{\acute{N}}{-1\choose I}{\acute{N}\choose I}F(P_\acute{n}-I\Delta_1P)}{\Delta_1P^\acute{n}};
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| \end{align}
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| </math>
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| : <math>
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| \begin{align}
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| \frac{d^\acute{n}F(P_0)}{dP^\acute{n}} & =\frac{d^{\acute{n}-1}F'(P_0)}{dP^{\acute{n}-1}}
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| =\frac{d^{\acute{n}-2}F''(P_0)}{dP^{\acute{n}-2}}
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| =\frac{d^{\acute{n}-3}F'''(P_0)}{dP^{\acute{n}-3}}=\cdots=\frac{d^{\acute{n}-r}F^{(r)}(P_0)}{dP^{\acute{n}-r}},
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| \\[10pt]
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| & =\frac{d^{\acute{n}-1}G(P_0)}{dP^{\acute{n}-1}} \\[10pt]
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| & =\frac{d^{\acute{n}-2}G'(P_0)}{dP^{\acute{n}-2}}=\ \frac{d^{\acute{n}-3}G''(P_0)}{dP^{\acute{n}-3}}=\cdots=\frac{d^{\acute{n}-r}G^{(r-1)}(P_0)}{dP^{\acute{n}-r}}, \\[10pt]
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| & {\color{white}.}\qquad\qquad\qquad=\frac{d^{\acute{n}-2}H(P_0)}{dP^{\acute{n}-2}}
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| =\ \frac{d^{\acute{n}-3}H'(P_0)}{dP^{\acute{n}-3}}=\cdots=\frac{d^{\acute{n}-r}H^{(r-2)}(P_0)}{dP^{\acute{n}-r}}, \\
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| & {\color{white}.}\qquad\qquad\qquad\qquad\qquad\qquad\ =\ \frac{d^{\acute{n}-3}I(P_0)}{dP^{\acute{n}-3}}
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| =\cdots=\frac{d^{\acute{n}-r}I^{(r-3)}(P_0)}{dP^{\acute{n}-r}}, \\[10pt]
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| & =F^{(\acute{n})}(P)=G^{(\acute{n}-1)}(P)=H^{(\acute{n}-2)}(P)=I^{(\acute{n}-3)}(P)=\cdots
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| \end{align}
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| </math>
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| : <math> | |
| \begin{align}
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| \frac{D^\acute{n}F(P_0)}{DP^\acute{n}} & =F[P_0,P_1,P_2,P_3,\ldots,P_{\acute{n}-3},P_{\acute{n}-2},P_{\acute{n}-1},P_\acute{n}], \\[10pt]
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| & =F^{(\acute{n})}(P_0 < P < P_\acute{n})=\sum_{TN=1}^{UT=\infty}\frac{F^{(\acute{n})}(P_{(tn)})}{UT}
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| \\[10pt]
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| & =F^{(\acute{n})}(LB < P < UB)=G^{(\acute{n}-1)}(LB < P < UB)= \cdots
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| \end{align}
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| </math>
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| | |
| ==Applying the divided difference==
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| The quintessential application of the divided difference is in the presentation of the definite integral, which is nothing more than a finite difference:
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| | |
| : <math> | |
| \begin{align}
| |
| \int_{LB}^{UB} G(p) \, dp & = \int_{LB}^{UB} F'(p) \, dp=F(UB)-F(LB), \\[10pt]
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| & =F[LB,UB]\Delta B, \\[10pt]
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| & =F'(LB < P < UB)\Delta B, \\[10pt]
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| & =\ G(LB < P < UB)\Delta B.
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| \end{align}
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| </math>
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| | |
| Given that the mean value, derivative expression form provides all of the same information as the classical integral notation, the mean value form may be the preferable expression, such as in writing venues that only support/accept standard [[ASCII]] text, or in cases that only require the average derivative (such as when finding the average radius in an elliptic integral).
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| This is especially true for definite integrals that technically have (e.g.) 0 and either <math>\pi\,\!</math> or <math>2\pi\,\!</math> as boundaries, with the same divided difference found as that with boundaries of 0 and <math>\begin{matrix}\frac{\pi}{2}\end{matrix}</math> (thus requiring less averaging effort):
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| | |
| : <math>
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| \begin{align}
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| \int_0^{2\pi} F'(p) \, dp & =4\int_0^{\frac{\pi}{2}} F'(p)\, dp=F(2\pi)-F(0)=4(F(\begin{matrix}\frac{\pi}{2}\end{matrix})-F(0)), \\[10pt]
| |
| & =2\pi F[0,2\pi]=2\pi F'(0 < P < 2\pi), \\[10pt]
| |
| & =2\pi F[0,\begin{matrix}\frac{\pi}{2}\end{matrix}] =2\pi F'(0 < P < \begin{matrix}\frac{\pi}{2}\end{matrix}).
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| \end{align}
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| </math>
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| This also becomes particularly useful when dealing with ''iterated'' and [[multiple integral|''multiple integral''s]] (ΔA = AU − AL, ΔB = BU − BL, ΔC = CU − CL):
| |
| | |
| : <math>
| |
| \begin{align}
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| & {} \qquad \int_{CL}^{CU}\int_{BL}^{BU} \int_{AL}^{AU} F'(r,q,p)\,dp\,dq\,dr \\[10pt]
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| & =\sum_{T\!C=1}^{U\!C=\infty}\left(\sum_{T\!B=1}^{U\!B=\infty}
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| \left(\sum_{T\!A=1}^{U\!A=\infty}F^{'}(R_{(tc)}:Q_{(tb)}:P_{(ta)})\frac{\Delta A}{U\!A}\right)\frac{\Delta B}{U\!B}\right)\frac{\Delta C}{U\!C}, \\[10pt]
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| & = F'(C\!L < R < CU:BL < Q < BU:AL < P <\!AU)
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| \Delta A\,\Delta B\,\Delta C.
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| \end{align}
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| </math>
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| | |
| Hence,
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| : <math>F'(R,Q:AL < P < AU)=\sum_{T\!A=1}^{U\!A=\infty}
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| \frac{F'(R,Q:P_{(ta)})}{U\!A};\,\!</math>
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| | |
| and
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| :<math>F'(R:BL < Q < BU:AL < P < AU)=\sum_{T\!B=1}^{U\!B=\infty}\left(\sum_{T\!A=1}^{U\!A=\infty}\frac{F'(R:Q_{(tb)}:P_{(ta)})}{U\!A}\right)\frac{1}{U\!B}.\,\!</math>
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| ==See also==
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| *[[Newton polynomial]]
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| *[[Rectangle method]]
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| *[[Quotient rule]]
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| ==External links==
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| *[http://cis.stvincent.edu/carlsond/ma109/diffquot.html Saint Vincent College: Br. David Carlson, O.S.B.—''MA109 The Difference Quotient'']
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| *[http://web.mat.bham.ac.uk/D.F.M.Hermans/msmxg6/ln/lnotes78.html University of Birmingham: Dirk Hermans—''Divided Differences'']
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| *Mathworld:
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| **[http://mathworld.wolfram.com/DividedDifference.html ''Divided Difference'']
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| **[http://mathworld.wolfram.com/Mean-ValueTheorem.html ''Mean-Value Theorem'']
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| *[http://www.cs.wisc.edu/wpis/abstracts/tr1415r.abs.html University of Wisconsin: [[Thomas W. Reps]] and Louis B. Rall—''Computational Divided Differencing and Divided-Difference Arithmetics'']
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| *[http://www.physics.arizona.edu/~restrepo/475A/Notes/sourcea/node31.html University of Arizona: Juan M. Restrepo—''Divided Differences'']
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| [[Category:Differential calculus]]
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| [[Category:Numerical analysis]]
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