Finite difference

Definition

Given a function $f$, a finite difference for $f$ with parameters real numbers $a$ and $b$ is the function:

$x\mapsto f(x+b)-f(x+a)$

The quotient of this by the value $b-a$ is a difference quotient expression.

There are three main types of finite differences parametrized by a positive real number $h$ of the corresponding difference quotient

Name	Symbol	Expression	Value of $a$	Value of $b$	Limit as $h\to 0^{+}$
forward difference	${\mathrm{\Delta }}_h[f](x)$	$f(x+h)-f(x)$	0	$h$	right-hand derivative $f{\text{'}}_{+}(x)$. If $f$ is differentiable at $x$, then this equals $f^{\prime }(x)$.
backward difference	${\mathrm{\nabla }}_h[f](x)$	$f(x)-f(x-h)$	$-h$	0	left-hand derivative $f{\text{'}}_{-}(x)$. If $f$ is differentiable at $x$, then this equals $f^{\prime }(x)$.
central difference	${\delta }_h[f](x)$	$f(x+\frac{h}{2})-f(x-\frac{h}{2})$	$\frac{-h}{2}$	$\frac{h}{2}$	If $f$ is differentiable at $x$, then this equals $f^{\prime }(x)$.

See also

• Higher-order finite difference