Difference Quotient

Difference Quotient

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:

• Forward difference: ΔF(P) = F(P + ΔP) - F(P);
• Central difference: δF(P) = F(P + ½ΔP) - F(P - ½ΔP);
• 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——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):

If ΔP is infinitesimal, then the difference quotient is a derivative, otherwise it is a divided difference: