The primary vehicle of calculus and other higher mathematics is the 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 (ΔP), as is the difference in their function result, the particular notation being determined by the direction of formation: For other uses, see Calculus (disambiguation). ... Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A... Look up Δ, δ in Wiktionary, the free dictionary. ...

• Forward difference:  ΔF(P) = F(P + ΔP) - F(P);
• Central difference:  δF(P) = F(P + 0.5ΔP)- F(P - 0.5ΔP);
• 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—iota—usually expressed in standard analysis as a limit: ), 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 the difference quotient (attributed to Isaac Newton), it is also known as Newton's quotient): Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. ... Sir Isaac Newton FRS (4 January 1643 – 31 March 1727) [ OS: 25 December 1642 – 20 March 1727][1] was an English physicist, mathematician, astronomer, natural philosopher, and alchemist. ...

If ΔP is infinitesimal, then the difference quotient is a derivative (which, when the object of integration, is known as an integrand), otherwise it is a divided difference: For a non-technical overview of the subject, see Calculus. ... In mathematics divided differences is a recursive division process. ...

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 ± (.5)ΔP (depending on the orientation—ΔF(P), δF(P) or ∇F(P)):

LB = Lower Boundary;   UB = Upper Boundary;

Anyone familiar with derivatives knows that they 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.
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, the nth point, equaling the degree/order:

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

 ΔP = Δ1P = P1 - P0 = P2 - P1 = P3 - P2 = ... = Pń - Pń-1; 

 ΔB = UB - LB = Pń - P0 = ΔńP = ŃΔ1P. 

The primary difference quotient (Ń = 1)

As a derivative

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), the Leibniz notation and derivative expressions do not distinguish P to P₀ or P_ń:

There are other derivative notations, but these are the most recognized, standard designations. In calculus, the Leibniz notation, named in honor of the 17th century German philosopher and mathematician Gottfried Wilhelm Leibniz (pronounced LIBE nits) was originally the use of dx and dy and so forth to represent infinitely small increments of quantities x and y, just as Δx and Δy represent finite... For a non-technical overview of the subject, see Calculus. ...

As a divided difference

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

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 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_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,

which links the mean value result with the divided difference:

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

In calculus, the mean value theorem states, roughly, that given a section of a smooth curve, there is a point on that section at which the derivative (slope) of the curve is equal (parallel) to the average derivative of the section. ...

Higher order difference quotients

Second order

Third order

Ńth Order

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:

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 or as boundaries, with the same divided difference found as that with boundaries of 0 and (thus requiring less averaging effort): Image:ASCII fullsvg There are 95 printable ASCII characters, numbered 32 to 126. ...

This also becomes particularly useful when dealing with iterated and multiple integrals (ΔA = AU - AL, ΔB = BU - BL, ΔC = CU - CL): To meet Wikipedias quality standards, this article or section may require cleanup. ...

Hence,

and

See also

• Newton polynomial
• Rectangle method
• Derivative
• Quotient rule

In the mathematical field of numerical analysis, a Newton polynomial, named after its inventor Isaac Newton, is the interpolation polynomial for a given set of data points in the Newton form. ... In mathematics, the rectangle method of integral calculus uses an approximation to a definite integral, made by finding the area of a series of rectangles. ... For a non-technical overview of the subject, see Calculus. ... In calculus, the quotient rule is a method of finding the derivative of a function that is the quotient of two other functions for which derivatives exist. ...

External links

• Saint Vincent College: Br. David Carlson, O.S.B.—MA109 The Difference Quotient

• University of Birmingham: Dirk Hermans—Divided Differences

• Mathworld:
  • Divided Difference
  • Mean-Value Theorem

• University of Wisconsin: Thomas W. Reps and Louis B. Rall—Computational Divided Differencing and Divided-Difference Arithmetics

• University of Arizona: Juan M. Restrepo—Divided Differences