Motivation
To calculate the derivative of the function at x, both approaches agree on the algebraic manipulations:
This is a non-standard computation using the hyperreals if we interpret Δx as an infinitesimal and let the symbol "" be the relation being infinitely close.
In order to make f ' a real-valued function, we must dispense with the final term . In the standard approach using only real numbers, that is done by taking the limit as tends to zero. In the non-standard approach using hyperreal numbers, the quantity is taken to be an infinitesimal, a nonzero number that is closer to 0 than to any nonzero real. The manipulations displayed above then show that is infinitely close to 2x, so the derivative of f at x is then 2x.
Discarding the "error term" is accomplished by an application of the standard part function. Dispensing with infinitesimal error terms was historically considered paradoxical by some writers, most notably George Berkeley.
Once the hyperreal number system (an infinitesimal-enriched continuum) is in place, one has successfully incorporated a large part of the technical difficulties at the foundational level. Thus, the epsilon, delta techniques that some believe to be the essence of analysis can be implemented once and for all at the foundational level, and the students needn't be "dressed to perform multiple-quantifier logical stunts on pretense of being taught infinitesimal calculus", to quote a recent study. More specifically, the basic concepts of calculus such as continuity, derivative, and integral can be defined using infinitesimals without reference to epsilon, delta. Thus, Keisler's Elementary Calculus: An Infinitesimal Approach defines continuity on page 125 in terms of infinitesimals, to the exclusion of epsilon, delta methods; similarly, the derivative is defined on page 45 using infinitesimals rather than an epsilon-delta approach; finally, integral is defined on page 183 in terms of infinitesimals, while epsilon, delta definitions are not introduced until page 282.
Read more about this topic: Non-standard Calculus
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