View source for Difference quotient

{{Short description|Expression in calculus}}
{{broader|Finite difference}}
In single-variable [[calculus]], the '''difference quotient''' is usually the name for the expression

:<math> \frac{f(x+h) - f(x)}{h} </math>

which when taken to the [[Limit of a function|limit]] as ''h'' approaches 0 gives the [[derivative]] of the [[Function (mathematics)|function]] ''f''.<ref name="LaxTerrell2013">{{cite book|author1=Peter D. Lax|author2=Maria Shea Terrell|title=Calculus With Applications|year=2013|publisher=Springer|isbn=978-1-4614-7946-8|page=119}}</ref><ref name="HockettBock2005">{{cite book|author1=Shirley O. Hockett|author2=David Bock|title=Barron's how to Prepare for the AP Calculus|year=2005|publisher=Barron's Educational Series|isbn=978-0-7641-2382-5|page=[https://archive.org/details/isbn_9780764177668/page/44 44]|url-access=registration|url=https://archive.org/details/isbn_9780764177668/page/44}}</ref><ref name="Ryan2010">{{cite book|author=Mark Ryan|title=Calculus Essentials For Dummies|year=2010|publisher=John Wiley & Sons|isbn=978-0-470-64269-6|pages=41–47}}</ref><ref name="NealGustafson2012">{{cite book|author1=Karla Neal|author2=R. Gustafson|author3=Jeff Hughes|title=Precalculus|year=2012|publisher=Cengage Learning|isbn=978-0-495-82662-0|page=133}}</ref> The name of the expression stems from the fact that it is the [[quotient]] of the [[Difference (mathematics)|difference]] of values of the function by the difference of the corresponding values of its argument (the latter is (''x'' + ''h'') - ''x'' = ''h'' in this case).<ref name="Comenetz2002">{{cite book|author=Michael Comenetz|title=Calculus: The Elements|year=2002|publisher=World Scientific|isbn=978-981-02-4904-5|pages=71–76 and 151–161}}</ref><ref name="Pasch2010">{{cite book|author=Moritz Pasch|title=Essays on the Foundations of Mathematics by Moritz Pasch|year=2010|publisher=Springer|isbn=978-90-481-9416-2|page=157}}</ref> The difference quotient is a measure of the [[average]] [[rate of change (mathematics)|rate of change]] of the function over an [[Interval (mathematics)|interval]] (in this case, an interval of length ''h'').<ref name="WilsonAdamson2008">{{cite book|author1=Frank C. Wilson|author2=Scott Adamson|title=Applied Calculus|year=2008|publisher=Cengage Learning|isbn=978-0-618-61104-1|page=177}}</ref><ref name="RubySellers2014"/>{{rp|237}}<ref name="HungerfordShaw2008">{{cite book|author1=Thomas Hungerford|author2=Douglas Shaw|title=Contemporary Precalculus: A Graphing Approach|year=2008|publisher=Cengage Learning|isbn=978-0-495-10833-7|pages=211–212}}</ref> The limit of the difference quotient (i.e., the derivative) is thus the [[instantaneous]] rate of change.<ref name="HungerfordShaw2008"/>

By a slight change in notation (and viewpoint), for an interval [''a'', ''b''], the difference quotient

:<math> \frac{f(b) - f(a)}{b-a}</math>

is called<ref name="Comenetz2002"/> the mean (or average) value of the derivative of ''f'' over the interval [''a'', ''b'']. This name is justified by the [[mean value theorem]], which states that for a [[differentiable function]] ''f'', its derivative ''{{prime|f}}'' reaches its [[Mean of a function|mean value]] at some point in the interval.<ref name="Comenetz2002"/> Geometrically, this difference quotient measures the [[slope]] of the [[secant line]] passing through the points with coordinates (''a'', ''f''(''a'')) and  (''b'', ''f''(''b'')).<ref name="Krantz2014">{{cite book|author=Steven G. Krantz|title=Foundations of Analysis|year=2014|publisher=CRC Press|isbn=978-1-4822-2075-9|page=127}}</ref>

Difference quotients are used as approximations in [[numerical differentiation]],<ref name="RubySellers2014">{{cite book|author1=Tamara Lefcourt Ruby|author2=James Sellers|author3=Lisa Korf |author4=Jeremy Van Horn |author5=Mike Munn|title=Kaplan AP Calculus AB & BC 2015|year=2014|publisher=Kaplan Publishing|isbn=978-1-61865-686-5|page=299}}</ref> but they have also been subject of criticism in this application.<ref name="GriewankWalther2008">{{cite book|author1=Andreas Griewank|author2=Andrea Walther|author2-link=Andrea Walther|title=Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation, Second Edition|url=https://books.google.com/books?id=qMLUIsgCwvUC&pg=PA2|year=2008|publisher=SIAM|isbn=978-0-89871-659-7|pages=2–}}</ref>

Difference quotients may also find relevance in applications involving [[Temporal discretization |Time discretization]], where the width of the time step is used for the value of h.

The difference quotient is sometimes also called the '''Newton quotient'''<ref name="Krantz2014"/><ref name="Lang1968">{{cite book|author=Serge Lang|title=Analysis 1|url=https://archive.org/details/analysisi0000lang|url-access=registration|year=1968|publisher=Addison-Wesley Publishing Company|page=[https://archive.org/details/analysisi0000lang/page/56 56]|author-link=Serge Lang}}</ref><ref name="Hahn1994">{{cite book|author=Brian D. Hahn|title=Fortran 90 for Scientists and Engineers|year=1994|publisher=Elsevier|isbn=978-0-340-60034-4|page=276}}</ref><ref name="ClaphamNicholson2009">{{cite book|author1=Christopher Clapham|author2=James Nicholson|title=The Concise Oxford Dictionary of Mathematics|url=https://archive.org/details/conciseoxforddic00clap|url-access=limited|year=2009|publisher=Oxford University Press|isbn=978-0-19-157976-9|page=[https://archive.org/details/conciseoxforddic00clap/page/n312 313]}}</ref> (after [[Isaac Newton]]) or '''Fermat's difference quotient''' (after [[Pierre de Fermat]]).<ref>Donald C. Benson, ''A Smoother Pebble: Mathematical Explorations'', Oxford University Press, 2003, p. 176.</ref>

==Overview==
The typical notion of the difference quotient discussed above is a particular case of a more general concept. 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:
*Forward difference: Δ''F''(''P'') = ''F''(''P'' + Δ''P'') − ''F''(''P'');
*Central difference: δF(P) = F(P + {{sfrac|1|2}}ΔP) − F(P − {{sfrac|1|2}}ΔP);
*Backward difference: ∇F(P) = F(P) − F(P − ΔP).
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,

*If |ΔP| is ''finite'' (meaning measurable), then ΔF(P) is known as a '''[[finite difference]]''', with specific denotations of DP and DF(P);
*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").

The function difference divided by the point difference is known as "difference quotient":

:<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>

If ΔP is infinitesimal, then the difference quotient is a ''[[derivative]]'', otherwise it is a ''[[divided differences|divided difference]]'':

:<math> \text{If } |\Delta P| = \mathit{ \iota}: \quad \frac{\Delta F(P)}{\Delta P}=\frac{dF(P)}{dP}=F'(P)=G(P);\,\!</math>

:<math> \text{If } |\Delta P| > \mathit{ \iota}: \quad \frac{\Delta F(P)}{\Delta P}=\frac{DF(P)}{DP}=F[P,P+\Delta P].\,\!</math>

==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:
<!--Improperly formatted formulae-->
   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.

==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.

==Higher-order difference quotients==

===Second order===

: <math>
\begin{align}
\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]
& =\frac{\frac{F(P_2)-F(P_1)}{\Delta_1P}-\frac{F(P_1)-F(P_0)}{\Delta_1P}}{\Delta_1P}, \\[10pt]
& =\frac{F(P_2)-2F(P_1)+F(P_0)}{\Delta_1P^2};
\end{align}
</math>

: <math>
\begin{align}
\frac{d^2F(P)}{dP^2} & = \frac{dF'(P)}{dP}=\frac{F'(P_1)-F'(P_0)}{dP}, \\[10pt]
& =\ \frac{dG(P)}{dP}=\frac{G(P_1)-G(P_0)}{dP}, \\[10pt]
& =\frac{F(P_2)-2F(P_1)+F(P_0)}{dP^2}, \\[10pt]
& =F''(P)=G'(P)=H(P)
\end{align}
</math>

: <math>
\begin{align}
\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]
& {\color{white}.} \qquad \ne\frac{F'(P_1)-F'(P_0)}{P_1-P_0}, \\[10pt]
& =F[P_0,P_1,P_2]=\frac{F(P_2)-2F(P_1)+F(P_0)}{(P_1-P_0)^2}, \\[10pt]
& =F''(P_0 < P < P_2)=\sum_{TN=1}^\infty \frac{F''(P_{(tn)})}{UT}, \\[10pt]
& =G'(P_0 < P < P_2)=H(P_0 < P < P_2).
\end{align}
</math>

===Third order===

: <math>
\begin{align}
\frac{\Delta^3F(P_0)}{\Delta_1P^3} & = \frac{\Delta^2 F'(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]
& =\frac{\frac{\frac{\Delta F(P_2)}{\Delta_1P}-\frac{\Delta F'(P_1)}{\Delta_1P}}{\Delta_1P}-
\frac{\frac{\Delta F'(P_1)}{\Delta_1P}-\frac{\Delta F'(P_0)}{\Delta_1P}}{\Delta_1P}}{\Delta_1P}, \\[10pt]
& =\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]
& =\frac{F(P_3)-3F(P_2)+3F(P_1)-F(P_0)}{\Delta_1P^3};
\end{align}
</math>

: <math>
\begin{align}
\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]
& =\frac{d^2G(P)}{dP^2}\ =\frac{dG'(P)}{dP}\ =\frac{G'(P_1)-G'(P_0)}{dP}, \\[10pt]
& {\color{white}.}\qquad\qquad\ \ =\frac{dH(P)}{dP}\ =\frac{H(P_1)-H(P_0)}{dP}, \\[10pt]
& =\frac{G(P_2)-2G(P_1)+G(P_0)}{dP^2}, \\[10pt]
& =\frac{F(P_3)-3F(P_2)+3F(P_1)-F(P_0)}{dP^3}, \\[10pt]
& =F'''(P)=G''(P)=H'(P)=I(P);
\end{align}
</math>

: <math>
\begin{align}
\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]
& {\color{white}.}\qquad\qquad\qquad\qquad\qquad\ \ \ne\frac{F''(P_1)-F''(P_0)}{P_1-P_0}, \\[10pt]
& =\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]
& =\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]
& =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]
& =F'''(P_0 < P < P_3)=\sum_{TN=1}^{UT=\infty}\frac{F'''(P_{(tn)})}{UT}, \\[10pt]
& =G''(P_0 < P < P_3)\ =H'(P_0 < P < P_3)=I(P_0 < P < P_3).
\end{align}
</math>

===''N''th order===

: <math>
\begin{align}
\Delta^\acute{n}F(P_0) & =F^{(\acute{n}-1)}(P_1)-F^{(\acute{n}-1)}(P_0), \\[10pt]
& =\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]
& =\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]
& {\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]
& = \cdots
\end{align}
</math>

: <math>
\begin{align}
\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]
& \frac{\nabla^\acute{n}F(P_\acute{n})}{\Delta_1P^\acute{n}} \\[10pt]
& =\frac{\sum_{I=0}^{\acute{N}}{-1\choose I}{\acute{N}\choose I}F(P_\acute{n}-I\Delta_1P)}{\Delta_1P^\acute{n}};
\end{align}
</math>

: <math>
\begin{align}
\frac{d^\acute{n}F(P_0)}{dP^\acute{n}} & =\frac{d^{\acute{n}-1}F'(P_0)}{dP^{\acute{n}-1}}
=\frac{d^{\acute{n}-2}F''(P_0)}{dP^{\acute{n}-2}}
=\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}},
 \\[10pt]
& =\frac{d^{\acute{n}-1}G(P_0)}{dP^{\acute{n}-1}} \\[10pt]
& =\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]
& {\color{white}.}\qquad\qquad\qquad=\frac{d^{\acute{n}-2}H(P_0)}{dP^{\acute{n}-2}}
=\ \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}}, \\
& {\color{white}.}\qquad\qquad\qquad\qquad\qquad\qquad\ =\ \frac{d^{\acute{n}-3}I(P_0)}{dP^{\acute{n}-3}}
=\cdots=\frac{d^{\acute{n}-r}I^{(r-3)}(P_0)}{dP^{\acute{n}-r}}, \\[10pt]
& =F^{(\acute{n})}(P)=G^{(\acute{n}-1)}(P)=H^{(\acute{n}-2)}(P)=I^{(\acute{n}-3)}(P)=\cdots
\end{align}
</math>

: <math>
\begin{align}
\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]
& =F^{(\acute{n})}(P_0 < P < P_\acute{n})=\sum_{TN=1}^{UT=\infty}\frac{F^{(\acute{n})}(P_{(tn)})}{UT}
 \\[10pt]
& =F^{(\acute{n})}(LB < P < UB)=G^{(\acute{n}-1)}(LB < P < UB)= \cdots 
\end{align}
</math>

==Applying the divided difference==
The quintessential application of the divided difference is in the presentation of the definite integral, which is nothing more than a finite difference:

: <math>
\begin{align}
\int_{LB}^{UB} G(p) \, dp & = \int_{LB}^{UB} F'(p) \, dp=F(UB)-F(LB), \\[10pt]
& =F[LB,UB]\Delta B, \\[10pt]
& =F'(LB < P < UB)\Delta B, \\[10pt]
& =\ G(LB < P < UB)\Delta B.
\end{align}
</math>

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).
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):

: <math>
\begin{align}
\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}).
\end{align}
</math>

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}
& {} \qquad \int_{CL}^{CU}\int_{BL}^{BU} \int_{AL}^{AU} F'(r,q,p)\,dp\,dq\,dr \\[10pt]
& =\sum_{T\!C=1}^{U\!C=\infty}\left(\sum_{T\!B=1}^{U\!B=\infty}
\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]
& = F'(C\!L < R < CU:BL < Q < BU:AL < P <\!AU)
\Delta A\,\Delta B\,\Delta C.
\end{align}
</math>

Hence,

: <math>F'(R,Q:AL < P < AU)=\sum_{T\!A=1}^{U\!A=\infty}
\frac{F'(R,Q:P_{(ta)})}{U\!A};\,\!</math>

and
:<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>

==See also==
*[[Divided differences]]
*[[Fermat theory]]
*[[Newton polynomial]]
*[[Rectangle method]]
*[[Quotient rule]]
*[[Symmetric difference quotient]]

==References==
{{Reflist|2}}

==External links==
*[http://cis.stvincent.edu/carlsond/ma109/diffquot.html Saint Vincent College: Br. David Carlson, O.S.B.—''MA109 The Difference Quotient''] {{Webarchive|url=https://web.archive.org/web/20050912183919/http://cis.stvincent.edu/carlsond/ma109/diffquot.html |date=2005-09-12 }}
*[http://web.mat.bham.ac.uk/D.F.M.Hermans/msmxg6/ln/lnotes78.html University of Birmingham: Dirk Hermans—''Divided Differences'']
*Mathworld:
**[http://mathworld.wolfram.com/DividedDifference.html  ''Divided Difference'']
**[http://mathworld.wolfram.com/Mean-ValueTheorem.html ''Mean-Value Theorem'']
*University of Wisconsin: [[Thomas W. Reps]] and Louis B. Rall — [http://www.cs.wisc.edu/wpis/abstracts/tr1415r.abs.html ''Computational Divided Differencing and Divided-Difference Arithmetics'']
*[http://giraldi.org/derivata/derivata.html Interactive simulator on difference quotient to explain the derivative]

{{Isaac Newton}}
{{Authority control}}

[[Category:Differential calculus]]
[[Category:Numerical analysis]]