Difference quotient

A difference quotient measures a function’s average rate of change across a finite interval by dividing the change in output by the change in input.

Average rate of change: The change in a quantity divided by the corresponding change in another quantity over an interval. A difference quotient is the function-based calculation of this rate.
Function: A rule assigning each input in its domain exactly one output. The quotient compares two outputs produced by the same function.
Numerical differentiation: The approximation of derivatives using function values at finitely many points. Finite difference quotients provide the basic derivative estimates.
Instantaneous rate of change: A function’s rate of change at a single input, defined through a limiting process when it exists. It is the limiting counterpart to the quotient’s finite-interval rate.
Secant line: A line that intersects a curve at two or more points. Its slope equals the difference quotient between the two intersection inputs.
Function notation: A symbolic convention for writing a function’s inputs and outputs, such as f(x). Expressions like f(x+h) specify the output at a shifted input.
Finite difference method: A numerical technique that approximates differential equations with equations involving values on a discrete grid. It builds discrete derivative approximations from difference quotients.
Central difference: A derivative approximation using function values at equal distances on both sides of an input. Its symmetric sampling differs from the usual one-sided quotient.
Derivative: The instantaneous rate of change of a function, defined by a limit of difference quotients when that limit exists. Taking the interval length toward zero turns the quotient into a derivative.
Increment: A measured change in a variable, often denoted by a difference between final and initial values. The denominator is an input increment, while the numerator is an output increment.

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