Editing Numerical differentiation (section)

==Finite differences==
{{further|Finite difference}}

The simplest method is to use finite difference approximations.

A simple two-point estimation is to compute the slope of a nearby [[secant line]] through the points {{math|(''x'', ''f''(''x''))}} and {{math|(''x'' + ''h'', ''f''(''x'' + ''h''))}}.<ref>Richard L. Burden, J. Douglas Faires (2000), ''Numerical Analysis'', (7th Ed),  Brooks/Cole. {{isbn|0-534-38216-9}}.</ref> Choosing a small number {{mvar|h}}, {{mvar|h}} represents a small change in {{mvar|x}}, and it can be either positive or negative.  The slope of this line is
<math display="block">\frac{f(x + h) - f(x)}{h}.</math>
This expression is [[Isaac Newton|Newton]]'s [[difference quotient]] (also known as a first-order [[divided difference]]).

The slope of this secant line differs from the slope of the tangent line by an amount that is approximately proportional to {{mvar|h}}. As {{mvar|h}} approaches zero, the slope of the secant line approaches the slope of the tangent line. Therefore, the true '''derivative of {{math|''f''}} at {{mvar|x}}''' is the limit of the value of the difference quotient as the secant lines get closer and closer to being a tangent line:
<math display="block">f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}.</math>

Since immediately [[substitution (logic)|substituting]] 0 for {{mvar|h}} results in <math>\frac{0}{0}</math> [[indeterminate form]], calculating the derivative directly can be unintuitive.

Equivalently, the slope could be estimated by employing positions {{math|''x'' − ''h''}} and {{mvar|x}}.

Another two-point formula is to compute the slope of a nearby secant line through the points {{math|(''x'' − ''h'', ''f''(''x'' − ''h''))}} and {{math|(''x'' + ''h'', ''f''(''x'' + ''h''))}}. The slope of this line is
<math display="block">\frac{f(x + h) - f(x - h)}{2h}.</math>

This formula is known as the [[symmetric difference quotient]]. In this case the first-order errors cancel, so the slope of these secant lines differ from the slope of the tangent line by an amount that is approximately proportional to <math>h^2</math>. Hence for small values of {{mvar|h}} this is a more accurate approximation to the tangent line than the one-sided estimation. However, although the slope is being computed at {{mvar|x}}, the value of the function at {{mvar|x}} is not involved.

The estimation error is given by
<math display="block">R = \frac{-f^{(3)}(c)}{6} h^2,</math>
where <math>c</math> is some point between <math>x - h</math> and <math>x + h</math>.
This error does not include the [[rounding error]] due to numbers being represented and calculations being performed in limited precision.

The symmetric difference quotient is employed as the method of approximating the derivative in a number of calculators, including [[TI-82]], [[TI-83]], [[TI-84]], [[TI-85]], all of which use this method with {{math|1=''h'' = 0.001}}.<ref name="Merseth2003">{{cite book |author=Katherine Klippert Merseth |title=Windows on Teaching Math: Cases of Middle and Secondary Classrooms |url=https://archive.org/details/windowsonteachin00mers |url-access=limited |year=2003 |publisher=Teachers College Press |isbn=978-0-8077-4279-2 |page=[https://archive.org/details/windowsonteachin00mers/page/n60 34]}}</ref><ref name="RubySellers2014">{{cite book |author1=Tamara Lefcourt Ruby |author2=James Sellers |author3=Lisa Korf |author4=Jeremy Van Horn |author5=Mike Munn |title=Kaplan AP Calculus AB & BC 2015 |year=2014 |publisher=Kaplan Publishing |isbn=978-1-61865-686-5 |page=299}}</ref>