The Gateaux Derivative of Map over Division Ring

I consider differential of mapping $f$ of continuous division ring as linear mapping the most close to mapping $f$. Different expressions which correspond to known deffinition of derivative are supplementary. I explore the Gateaux derivative of higher order and Taylor series. The Taylor series allow solving of simple differential equations. As an example of solution of differential equation I considered a model of exponent. I considered application of obtained theorems to complex field and quaternion algebra. In contrast to complex field in quaternion algebra congugation is linear function of original number \bar a=a+iai+jaj+kak . In quaternion algebra this difference leads to the absence of analogue of the Cauchy Riemann equations that are well known in the theory of complex function.

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