Editing Hyperreal number (section)

=== Differentiation ===
One of the key uses of the hyperreal number system is to give a precise meaning to the differential operator ''d'' as used by Leibniz to define the derivative and the integral.

For any real-valued function <math>f,</math> the differential <math>df</math> is defined as a map which sends every ordered pair <math>(x,dx)</math> (where <math>x</math> is real and <math>dx</math> is nonzero infinitesimal) to an infinitesimal
: <math>df(x,dx) := \operatorname{st}\left(\frac{f(x + dx) - f(x)}{dx}\right) \ dx.</math>

Note that the very notation "<math>dx</math>" used to denote any infinitesimal is consistent with the above definition of the operator <math>d,</math> for if one interprets <math>x</math> (as is commonly done) to be the function <math>f(x)=x,</math> then for every <math>(x,dx)</math> the differential <math>d(x)</math> will equal the infinitesimal <math>dx</math>.

A real-valued function <math>f</math> is said to be differentiable at a point <math>x</math> if the quotient
: <math>\frac{df(x,dx)}{dx}=\operatorname{st}\left(\frac{f(x + dx) - f(x)}{dx}\right)</math>
is the same for all nonzero infinitesimals <math>dx.</math> If so, this quotient is called the derivative of <math>f</math> at <math>x</math>.

For example, to find the [[derivative]] of the [[function (mathematics)|function]] <math>f(x)=x^2</math>, let <math>dx</math> be a non-zero infinitesimal. Then,
: {|
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|<math>\frac{df(x,dx)}{dx}</math>
|<math>=\operatorname{st}\left(\frac{f(x + dx) - f(x)}{dx}\right)</math>
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|<math>=\operatorname{st}\left(\frac{x^2 + 2x \cdot dx + (dx)^2 -x^2}{dx}\right)</math>
|-
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|<math>=\operatorname{st}\left(\frac{2x \cdot dx +  (dx)^2}{dx}\right)</math>
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|<math>=\operatorname{st}\left(\frac{2x \cdot dx}{dx} + \frac{(dx)^2}{dx}\right)</math>
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|<math>=\operatorname{st}\left(2x + dx\right)</math>
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|<math>=2x</math>
|}

The use of the standard part in the definition of the derivative is a rigorous alternative to the traditional practice of neglecting the square {{Citation needed|reason=if dx got to zero in limit then dx square go there too and doing this faster, this is rigorous|date=February 2018}} of an infinitesimal quantity. [[Dual number]]s are a number system based on this idea. After the third line of the differentiation above, the typical method from Newton through the 19th century would have been simply to discard the ''dx''<sup>2</sup> term. In the hyperreal system,
''dx''<sup>2</sup>&nbsp;≠&nbsp;0, since ''dx'' is nonzero, and the transfer principle can be applied to the statement that the square of any nonzero number is nonzero. However, the quantity ''dx''<sup>2</sup>  is infinitesimally small compared to ''dx''; that is, the hyperreal system contains a hierarchy of infinitesimal quantities.

Using hyperreal numbers for differentiation allows for a more algebraically manipulable approach to derivatives.  In standard differentiation, partial differentials and higher-order differentials are not independently manipulable through algebraic techniques.  However, using the hyperreals, a system can be established for doing so, though resulting in a slightly different notation.<ref>{{cite book |last=Fite |first=Isabelle |title=Operator Theory - Recent Advances, New Perspectives and Applications |chapter=Total and Partial Differentials as Algebraically Manipulable Entities |arxiv=2210.07958 |date=2023|doi=10.5772/intechopen.107285 |isbn=978-1-83880-992-8 }}</ref>