There are two main trends of nonstandard analysis in which some form of infinitesimal quantities are defined: the hyperreal number system (established by A. Robinson and formalised also by K.D. Stroyan and W.A.J. Luxemburg) in which the real numbers are extended to contain infinitesimals which are not real numbers and IST (established by E. Nelson) an axiomatic extension in which it is the language that is extended to enable defining infinitely small real numbers.
Both approaches are interesting in their own right and share a common principle: there are standard numbers and nonstandard numbers. Both approaches have a difficulty with the derivative: there is a definition for the derivative of a standard function at a standard value and that formula can be made very intuitive. But to extend the definition to nonstandard values and nonstandard functions, the level of technicality required to do it rigorously seems to defeat the purpose.
If the derivative is not easily defined at nonstandard points, the derivative function becomes problematic. If nonstandard functions are not easily defined, the tangent function becomes problematic.
The distinction between internal and external properties is crucial in both approaches but it is technically sophisticated and requires a good understanding of mathematical logic.
One of the authors of the book faced these problems when a first attempt at teaching using infinitesimals was used in 2000, and in 2002 the meeting with K. Hrbacek lead to the present approach. It is an axiomatic extension similar to Nelson’s IST but not limited to two levels. The words used (ultrasmall and ultralarge) avoid confusion with other similar but not identical concepts.
In our theory (analysis with ultrasmall numbers), the definition of the derivative is valid for all numbers and properties that satisfy a specific syntactic rule are automatically internal. External properties are altogether avoided. It can be made intuitively acceptable to students and there is a certain number of teachers who have been using this approach in two Geneva High Schools for the last 15 years. The integration of these students in university courses (or polytechnical school EPFL) has been good.