Difference quotient

Definition

Algebraic definition

The difference quotient of a function between two distinct points in its domain is defined as the quotient of the difference between the function values at the two points by the difference between the two points.

In symbols, if $f$ is a function defined on some subset of the reals and $x_{1},x_{2}$ are distinct elements in the domain of $f$, then the difference quotient of $f$ between $x_{1}$ and $x_{2}$, denoted $\Delta f(x_{1},x_{2})$, is defined as:

$\!\Delta f(x_{1},x_{2}):={\frac {f(x_{2})-f(x_{1})}{x_{2}-x_{1}}}$

Note that the definition is symmetric in $x_{1}$ and $x_{2}$, i.e., we have:

$\!\Delta f(x_{1},x_{2})=\Delta f(x_{2},x_{1})$

Geometric definition

The difference quotient of a function between two distinct points in its domain is defined as the slope of the chord joining the corresponding points in the graph of the function.

In symbols, if $f$ is a function defined on some subset of the reals and $x_{1},x_{2}$ are distinct elements in the domain of $f$, then the difference quotient of $f$ between $x_{1}$ and $x_{2}$ is defined as the slope of the line segment joining the points $(x_{1},f(x_{1}))$ and $(x_{2},f(x_{2}))$, both of which are part of the graph of $f$.

Related notions

• Derivative is defined as a limit of the difference quotient as one point approaches the other.