Of course, infinitesimals are not real numbers. Together with the reals, they form the hyperreal numbers, which provide the framework for this new approach to analysis, pioneered by the Canadian Abraham Robinson (see Tangente 149).
Let's use a simple example to see how to calculate a derivative in nonstandard analysis. This is essentially the method Fermat used to find a maximum, recast in modern language. We wish to calculate the derivative of the function defined by f ( x ) = *x 2. Let x be an arbitrary real number and ε an infinitesimal. Consider f ( x + ε ), namely *x 2 + 2ε x +ε 2, a number infinitely close to *x 2: the two numbers are "almost equal" (Fermat wrote that there was an "adequality" between them). Their difference is infinitesimal. We then scale this difference by dividing it by ε; the quotient of these two infinitesimals is equal to 2 x + ε and is therefore infinitely close to 2 x. To obtain an equality, we take the unique real number infinitely close to this quotient: it is 2 x. Technically, this is called the standard part of the quotient.
Thus, provided it exists, the derivative of a function f at x is the standard part of
where ε denotes an infinitesimal.
