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 $h$

Name	Expression	Value of $a$	Value of $b$	Limit as $h\to 0^{+}$
forward difference	$f(x+h)-f(x)$	0	$h$	right-hand derivative $f'_{+}(x)$. If $f$ is differentiable at $x$, then this equals $f'(x)$.
backward difference	$f(x)-f(x-h)$	-h</math>	0	left-hand derivative $f'_{-}(x)$. If $f$ is differentiable at $x$, then this equals $f'(x)$.
central difference	$f(x+h)-f(x-h)$	$-h$	$h$	If $f$ is differentiable at $x$, then this equals $f'(x)$.

See also

• Higher-order finite difference