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Proceedings of the Royal Irish Academy, Section A |
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== The derivative for quaternions ==
Since the time of Hamilton, it has been realized that requiring the independence of the derivative from the path that a differential follows toward zero is too restrictive: it excludes even <math>f(q) = q^2</math> from differentiability. Therefore, a direction-dependent derivative is necessary for functions of a quaternion variable.<ref>{{harv|Hamilton|1866|loc=Chapter II, On differentials and developments of functions of quaternions, pp. 391–495}}</ref><ref>{{harv|Laisant|1881|loc=Chapitre 5: Différentiation des Quaternions, pp. 104–117}}</ref>
Considering of the increment of polynomial function of quaternionic argument shows that the increment is linear map of increment of the argument.
This statement is the basis for the following definition.
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</math>
Therefore, the differential of the map <math>f</math> may be expressed as
/math With Brackets on either side.
:<math>\frac{d f(x)}{d x}\circ dx=
\left(
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