Editing Difference quotient (section)

==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>