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