difference quotient

Meanings

1. A mathematical quantity of the form (f(a+h)-f(a))/h for some function f(x), having two main interpretations:
2. The average rate of change of the function f(x) over the interval [a,,a+h]. As h approaches zero, the value of (f(a+h)-f(a))/h approaches the derivative or instantaneous rate of change of f(x) at x=a.
3. The slope of the secant line passing through the points (a,,f(a)) and (a+h,,f(a+h)). As the horizontal distance h>0 between the points approaches zero, the slope of this secant line approaches the slope of the tangent line to f(x) at x=a, which is equal to the derivative of f(x) at x=a.