Definition of Derivative
The hyperreals can be constructed in the framework of Zermelo-Fraenkel set theory, the standard axiomatisation of set theory used elsewhere in mathematics. To give an intuitive idea for the hyperreal approach, note that, naively speaking, non-standard analysis postulates the existence of positive numbers ε which are infinitely small, meaning that ε is smaller than any standard positive real, yet greater than zero. Every real number x is surrounded by an infinitesimal "cloud" of hyperreal numbers infinitely close to it. To define the derivative of f at a standard real number x in this approach, one no longer needs an infinite limiting process as in standard calculus. Instead, one sets
where st is the standard part function, yielding the real number infinitely close to the hyperreal argument of st, and is the natural extension of to the hyperreals.
Read more about this topic: Non-standard Calculus
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