In single-variable calculus, the differencequotient is usually the name for the expression which when taken to the limit as h approaches 0 gives the derivative of the functionf. The name of the expression stems from the fact that it is the quotient of the difference of values of the function by the difference of the corresponding values of its argument. The difference quotient is a measure of the averagerate of change of the function over an interval. The limit of the difference quotient is thus the instantaneous rate of change. By a slight change in notation, for an interval , the difference quotient is called the mean value of the derivative of f over the interval . This name is justified by the mean value theorem, which states that for a differentiable functionf, its derivative f′ reaches its mean value at some point in the interval. Geometrically, this difference quotient measures the slope of the secant line passing through the points with coordinates and . Difference quotients are used as approximations in numerical differentiation, but they have also been subject of criticism in this application. The difference quotient is sometimes also called the Newton quotient or Fermat's difference quotient.
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. Its "input value" is its argument, usually a point expressible on a graph. The difference between two points, themselves, is known as their Delta, as is the difference in their function result, the particular notation being determined by the direction of formation:
Forward difference: ΔF = F − F;
Central difference: δF = F − F;
Backward difference: ∇F = F − F.
The general preference is the forward orientation, as F is the base, to which differences are added to it. Furthermore,
If |ΔP| is finite, then ΔF is known as a finite difference, with specific denotations of DP and DF;
If |ΔP| is infinitesimal, then ΔF is known as an infinitesimal difference, with specific denotations of dP and dF.
The function difference divided by the point difference is known as "difference quotient": If ΔP is infinitesimal, then the difference quotient is a derivative, otherwise it is a divided difference:
Regardless if ΔP is infinitesimal or finite, there is a point range, where the boundaries are P ± ΔP, δF or ∇F): Derivatives can be regarded as functions themselves, harboring their own derivatives. Thus each function is home to sequentialdegrees 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, where LB = P0 and UB = Pń, the nth point, equaling the degree/order: LB = P0 = P0 + 0Δ1P = Pń − Δ1P; P1 = P0 + 1Δ1P = Pń − Δ1P; P2 = P0 + 2Δ1P = Pń − Δ1P; P3 = P0 + 3Δ1P = Pń − Δ1P; ↓ ↓ ↓ ↓ Pń-3 = P0 + Δ1P = Pń − 3Δ1P; Pń-2 = P0 + Δ1P = Pń − 2Δ1P; Pń-1 = P0 + Δ1P = Pń − 1Δ1P; UB = Pń-0 = P0 + Δ1P = Pń − 0Δ1P = Pń; ΔP = Δ1P = P1 − P0 = P2 − P1 = P3 − P2 =... = Pń − Pń-1; ΔB = UB − LB = Pń − P0 = ΔńP = ŃΔ1P.
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. This is especially true for definite integrals that technically have 0 and either or as boundaries, with the same divided difference found as that with boundaries of 0 and : This also becomes particularly useful when dealing with iterated and multiple integrals : Hence, and
