Editing Dual number (section)

==Automatic differentiation==
{{anchor|Differentiation}}
One application of dual numbers is [[automatic differentiation]]. Any polynomial

:<math>P(x) = p_0 + p_1x + p_2x^2 + \cdots + p_nx^n</math>

with real coefficients can be extended to a function of a dual-number-valued argument,

:<math>\begin{align}
P(a + b\varepsilon)
&= p_0 + p_1(a + b\varepsilon) + \cdots + p_n(a + b\varepsilon)^n \\[2mu]
&= p_0 + p_1 a + p_2 a^2 + \cdots + p_n a^n + p_1 b\varepsilon + 2 p_2 a b\varepsilon + \cdots + n p_n a^{n-1} b\varepsilon \\[5mu]
&= P(a) + bP'(a)\varepsilon,
\end{align}</math>

where <math>P'</math> is the derivative of <math>P.</math>

More generally, any (analytic) real function can be extended to the dual numbers via its [[Taylor series]]:

:<math>f(a + b\varepsilon) = \sum_{n=0}^\infty \frac{f^{(n)} (a)b^n \varepsilon^n}{n!} = f(a) + bf'(a)\varepsilon,</math>

since all terms involving {{math|''ε''<sup>2</sup>}} or greater powers are trivially {{math|0}} by the definition of {{mvar|ε}}.

By computing compositions of these functions over the dual numbers and examining the coefficient of {{mvar|ε}} in the result we find we have automatically computed the derivative of the composition.

A similar method works for polynomials of {{mvar|n}} variables, using the [[exterior algebra]] of an {{mvar|n}}-dimensional vector space.