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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 differentiation. 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. From this, a definition can be made:
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