The primary vehicle of calculus and other higher mathematics is the function. Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of 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:

• Forward difference:  ΔF(P) = F(P + ΔP) - F(P);
• Central difference:  δF(P) = F(P + 0. Delta (uppercase Δ, lowercase δ; Δέλτα Thelta is the fourth letter of the Greek alphabet. 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"). Infinitesimals (from a 17th century Modern Latin coinage infinitesimus, originally referring to the " Infinite[[ th]]" member of a series have
  

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):

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:

Contents 1 Defining the point range 2 The primary difference quotient (Ń = 1) 2.1 As a derivative 2.2 As a divided difference 3 Higher order difference quotients 3.1 Second order 3.2 Third order 3.3 Ńth Order 4 Applying the divided difference 5 See also 6 External links

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 ± (. Sir Isaac Newton, FRS (ˈnjuːtən 4 January 1643 31 March 1727) Biography Early years See also Isaac Newton's early life and achievements In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change In Mathematics divided differences is a recursive division process 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, Leibniz's notation, named in honor of the 17th-century German philosopher and mathematician Gottfried Wilhelm Leibniz In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change

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. In Calculus, the mean value theorem states roughly that given a section of a smooth curve there is at least one point on that section at which the Derivative

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.

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). American Standard Code for Information Interchange ( ASCII) 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):

This also becomes particularly useful when dealing with iterated and multiple integrals (ΔA = AU - AL, ΔB = BU - BL, ΔC = CU - CL):

Hence,

   

and

See also

• Newton polynomial
• Rectangle method
• Quotient rule

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

The multiple integral is a type of definite Integral extended to functions of more than one real Variable, for example In the Mathematical field of Numerical analysis, a Newton polynomial, named after its inventor Isaac Newton, is the interpolation Polynomial In Mathematics, specifically in Integral calculus, the rectangle method (also called the Midpoint or Mid-Ordinate Rule) uses an Approximation 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

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