difference quotient
Noun
Meaning
difference quotient (plural difference quotients)
(mathematical analysis) The slope of a secant as obtained by dividing the vertical difference between two intersections of the secant and a curve with their horizontal distance. When the horizontal distance approaches zero the slope of the secant approaches the derivative of the curve.
Examples
More directly, given a small change in x, the relative change in x is while the relative change in is Taking the ratio yields: : The last term is the difference quotient (the slope of the secant line), and taking the limit yields the derivative.
Consequently, the secant lines do not approach any single slope, so the limit of the difference quotient does not exist.
However, the usual difference quotient does not make sense in higher dimensions because it is not usually possible to divide vectors.
Rules of computation main The derivative of a function can, in principle, be computed from the definition by considering the difference quotient, and computing its limit.