Difference quotient

For broader coverage of this topic, seeFinite difference.

In single-variablecalculus, thedifference quotient is usually the name for the expression

{\frac {f(x+h)-f(x)}{h}}

which when taken to thelimit ash approaches 0 gives thederivative of thefunctionf.^[1]^[2]^[3]^[4] The name of the expression stems from the fact that it is thequotient of thedifference 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).^[5]^[6] The difference quotient is a measure of theaverage rate of change of the function over aninterval (in this case, an interval of lengthh).^[7]^[8]^: 237^[9] The limit of the difference quotient (i.e., the derivative) is thus theinstantaneous rate of change.^[9]

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

{\frac {f(b)-f(a)}{b-a}}

is called^[5] the mean (or average) value of the derivative off over the interval [a,b]. This name is justified by themean value theorem, which states that for adifferentiable functionf, its derivativef′ reaches itsmean value at some point in the interval.^[5] Geometrically, this difference quotient measures theslope of thesecant line passing through the points with coordinates (a,f(a)) and (b,f(b)).^[10]

Difference quotients are used as approximations innumerical differentiation,^[8] but they have also been subject of criticism in this application.^[11]

Difference quotients may also find relevance in applications involvingTime discretization, where the width of the time step is used for the value of h.

The difference quotient is sometimes also called theNewton quotient^[10]^[12]^[13]^[14] (afterIsaac Newton) orFermat's difference quotient (afterPierre de Fermat).^[15]

Overview

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The typical notion of the difference quotient discussed above is a particular case of a more general concept. The primary vehicle ofcalculus and other higher mathematics is thefunction. Its "input value" is itsargument, usually a point ("P") expressible on a graph. The difference between two points, themselves, is known as theirDelta (ΔP), as is the difference in their function result, the particular notation being determined by the direction of formation:

Forward difference: ΔF(P) =F(P + ΔP) −F(P);
Central difference: δF(P) = F(P +⁠1/2⁠ΔP) − F(P −⁠1/2⁠ΔP);
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| isfinite (meaning measurable), then ΔF(P) is known as afinite difference, with specific denotations of DP and DF(P);
If |ΔP| isinfinitesimal (an infinitely small amount— $\iota$ —usually expressed in standard analysis as a limit: $\lim _{\Delta P\rightarrow 0}\,\!$ ), then ΔF(P) is known as aninfinitesimal 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":

{\frac {\Delta F(P)}{\Delta P}}={\frac {F(P+\Delta P)-F(P)}{\Delta P}}={\frac {\nabla F(P+\Delta P)}{\Delta P}}.\,\!

If ΔP is infinitesimal, then the difference quotient is aderivative, otherwise it is adivided difference:

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

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

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)):

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, ordifferentiation. This property can be generalized to all difference quotients.
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_i), where LB =P₀ and UB =P_ń, thenth point, equaling the degree/order:

  LB =  P₀  = P₀ + 0Δ₁P     = P_ń − (Ń-0)Δ₁P;        P₁  = P₀ + 1Δ₁P     = P_ń − (Ń-1)Δ₁P;        P₂  = P₀ + 2Δ₁P     = P_ń − (Ń-2)Δ₁P;        P₃  = P₀ + 3Δ₁P     = P_ń − (Ń-3)Δ₁P;            ↓      ↓        ↓       ↓       P_ń-3 = P₀ + (Ń-3)Δ₁P = P_ń − 3Δ₁P;       P_ń-2 = P₀ + (Ń-2)Δ₁P = P_ń − 2Δ₁P;       P_ń-1 = P₀ + (Ń-1)Δ₁P = P_ń − 1Δ₁P;  UB = P_ń-0 = P₀ + (Ń-0)Δ₁P = P_ń − 0Δ₁P = P_ń;

  ΔP = Δ₁P = P₁ − P₀ = P₂ − P₁ = P₃ − P₂ = ... = P_ń − P_ń-1;

  ΔB = UB − LB = P_ń − P₀ = Δ_ńP = ŃΔ₁P.

The primary difference quotient (Ń = 1)

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{\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 _{1}P}}={\frac {F(P_{1})-F(P_{0})}{P_{1}-P_{0}}}.\,\!

As a derivative

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The difference quotient as a derivative needs no explanation, other than to point out that, since P₀ essentially equals P₁ = P₂ = ... = P_ń (as the differences are infinitesimal), theLeibniz notation and derivative expressions do not distinguish P to P₀ or P_ń:

{\frac {dF(P)}{dP}}={\frac {F(P_{1})-F(P_{0})}{dP}}=F'(P)=G(P).\,\!

There areother derivative notations, but these are the most recognized, standard designations.

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:

{\begin{aligned}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{aligned}}

In this interpretation, P_ã 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_ã is found in themean value theorem of calculus, which says:

For any function that is continuous on [LB,UB] and differentiable on (LB,UB) there exists some P_ã in the interval (LB,UB) such that the secant joining the endpoints of the interval [LB,UB] is parallel to the tangent at P_ã.

Essentially, P_ã denotes some value of P between LB and UB—hence,

P_{\tilde {a}}:=LB<P<UB=P_{0}<P<P_{\acute {n}}\,\!

which links the mean value result with the divided difference:

{\begin{aligned}{\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{aligned}}

As there is, by its very definition, a tangible difference between LB/P₀ and UB/P_ń, the Leibniz and derivative expressionsdo requiredivarication of the function argument.

Higher-order difference quotients

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Second order

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{\begin{aligned}{\frac {\Delta ^{2}F(P_{0})}{\Delta _{1}P^{2}}}&={\frac {\Delta F'(P_{0})}{\Delta _{1}P}}={\frac {{\frac {\Delta F(P_{1})}{\Delta _{1}P}}-{\frac {\Delta F(P_{0})}{\Delta _{1}P}}}{\Delta _{1}P}},\\[10pt]&={\frac {{\frac {F(P_{2})-F(P_{1})}{\Delta _{1}P}}-{\frac {F(P_{1})-F(P_{0})}{\Delta _{1}P}}}{\Delta _{1}P}},\\[10pt]&={\frac {F(P_{2})-2F(P_{1})+F(P_{0})}{\Delta _{1}P^{2}}};\end{aligned}}

{\begin{aligned}{\frac {d^{2}F(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{aligned}}

{\begin{aligned}{\frac {D^{2}F(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 \neq {\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{aligned}}

Third order

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{\begin{aligned}{\frac {\Delta ^{3}F(P_{0})}{\Delta _{1}P^{3}}}&={\frac {\Delta ^{2}F'(P_{0})}{\Delta _{1}P^{2}}}={\frac {\Delta F''(P_{0})}{\Delta _{1}P}}={\frac {{\frac {\Delta F'(P_{1})}{\Delta _{1}P}}-{\frac {\Delta F'(P_{0})}{\Delta _{1}P}}}{\Delta _{1}P}},\\[10pt]&={\frac {{\frac {{\frac {\Delta F(P_{2})}{\Delta _{1}P}}-{\frac {\Delta F'(P_{1})}{\Delta _{1}P}}}{\Delta _{1}P}}-{\frac {{\frac {\Delta F'(P_{1})}{\Delta _{1}P}}-{\frac {\Delta F'(P_{0})}{\Delta _{1}P}}}{\Delta _{1}P}}}{\Delta _{1}P}},\\[10pt]&={\frac {{\frac {F(P_{3})-2F(P_{2})+F(P_{1})}{\Delta _{1}P^{2}}}-{\frac {F(P_{2})-2F(P_{1})+F(P_{0})}{\Delta _{1}P^{2}}}}{\Delta _{1}P}},\\[10pt]&={\frac {F(P_{3})-3F(P_{2})+3F(P_{1})-F(P_{0})}{\Delta _{1}P^{3}}};\end{aligned}}

{\begin{aligned}{\frac {d^{3}F(P)}{dP^{3}}}&={\frac {d^{2}F'(P)}{dP^{2}}}={\frac {dF''(P)}{dP}}={\frac {F''(P_{1})-F''(P_{0})}{dP}},\\[10pt]&={\frac {d^{2}G(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{aligned}}

{\begin{aligned}{\frac {D^{3}F(P_{0})}{DP^{3}}}&={\frac {D^{2}F'(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 \ \ \neq {\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{aligned}}

Nth order

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{\begin{aligned}\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 _{1}P}}-{\frac {F^{({\acute {n}}-2)}(P_{1})-F^{({\acute {n}}-2)}(P_{0})}{\Delta _{1}P}},\\[10pt]&={\frac {{\frac {F^{({\acute {n}}-3)}(P_{3})-F^{({\acute {n}}-3)}(P_{2})}{\Delta _{1}P}}-{\frac {F^{({\acute {n}}-3)}(P_{2})-F^{({\acute {n}}-3)}(P_{1})}{\Delta _{1}P}}}{\Delta _{1}P}}\\[10pt]&{\color {white}.}\qquad -{\frac {{\frac {F^{({\acute {n}}-3)}(P_{2})-F^{({\acute {n}}-3)}(P_{1})}{\Delta _{1}P}}-{\frac {F^{({\acute {n}}-3)}(P_{1})-F^{({\acute {n}}-3)}(P_{0})}{\Delta _{1}P}}}{\Delta _{1}P}},\\[10pt]&=\cdots \end{aligned}}

{\begin{aligned}{\frac {\Delta ^{\acute {n}}F(P_{0})}{\Delta _{1}P^{\acute {n}}}}&={\frac {\sum _{I=0}^{\acute {N}}{-1 \choose {\acute {N}}-I}{{\acute {N}} \choose I}F(P_{0}+I\Delta _{1}P)}{\Delta _{1}P^{\acute {n}}}};\\[10pt]&{\frac {\nabla ^{\acute {n}}F(P_{\acute {n}})}{\Delta _{1}P^{\acute {n}}}}\\[10pt]&={\frac {\sum _{I=0}^{\acute {N}}{-1 \choose I}{{\acute {N}} \choose I}F(P_{\acute {n}}-I\Delta _{1}P)}{\Delta _{1}P^{\acute {n}}}};\end{aligned}}

{\begin{aligned}{\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{aligned}}

{\begin{aligned}{\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{aligned}}

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:

{\begin{aligned}\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{aligned}}

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 standardASCII 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 $\pi \,\!$ or $2\pi \,\!$ as boundaries, with the same divided difference found as that with boundaries of 0 and ${\begin{matrix}{\frac {\pi }{2}}\end{matrix}}$ (thus requiring less averaging effort):

{\begin{aligned}\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{aligned}}

This also becomes particularly useful when dealing withiterated andmultiple integrals (ΔA = AU − AL, ΔB = BU − BL, ΔC = CU − CL):

{\begin{aligned}&{}\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{aligned}}

Hence,

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

and

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}}.\,\!

References

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