Editing Finite difference (section)

== Relation with derivatives ==
{{main|difference quotient}}
{{anchor|finite difference approximation}}
The approximation of [[derivative]]s by finite differences plays a central role in [[finite difference method]]s for the [[numerical analysis|numerical]] solution of [[differential equation]]s, especially [[boundary value problem]]s.

The [[derivative]] of a function {{mvar|f}} at a point {{mvar|x}}  is defined by the [[limit of a function|limit]]
<math display="block"> f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}. </math>

If {{mvar|h}} has a fixed (non-zero) value instead of approaching zero, then the right-hand side of the above equation would be written
<math display="block"> \frac{f(x + h) - f(x)}{h} = \frac{\Delta_h[f](x)}{h}. </math>

Hence, the forward difference divided by {{mvar|h}} approximates the derivative when {{mvar|h}} is small. The error in this approximation can be derived from [[Taylor's theorem]]. Assuming that {{mvar|f}} is twice differentiable, we have
<math display="block"> \frac{\Delta_h[f](x)}{h} - f'(x) = o(h)\to 0 \quad  \text{as }h \to 0. </math>

The same formula holds for the backward difference:
<math display="block"> \frac{\nabla_h[f](x)}{h} - f'(x) = o(h)\to 0 \quad  \text{as }h \to 0. </math>

However, the central (also called centered) difference yields a more accurate approximation. If {{mvar|f}} is three times differentiable,
<math display="block"> \frac{\delta_h[f](x)}{h} - f'(x) =  o\left(h^2\right) . </math>

The main problem{{citation needed|date=December 2017}} with the central difference method, however, is that oscillating functions can yield zero derivative. If  {{math|1=''f''(''nh'') = 1}} for {{mvar|n}} odd, and {{math|1=''f''(''nh'') = 2}}  for {{mvar|n}} even, then {{math|1=''f''′(''nh'') = 0}} if it is calculated with the [[central difference scheme]]. This is particularly troublesome if the domain of {{mvar|f}} is discrete. See also [[Symmetric derivative]].

Authors for whom finite differences mean finite difference approximations define the forward/backward/central differences as the quotients given in this section (instead of employing the definitions given in the previous section).<ref name="WilmottHowison1995"/><ref name="Olver2013"/><ref name="Chaudhry2007"/>