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Infinitesimal
Infinitesimals are objects too small to measure.
Infinitesimal calculus
Newton and Leibniz developed the calculus based on an intuitive notion of an infinitesimal. In 1870 Karl Weierstrass provided the first rigorous treatment of the calculus, using the limit method. But in 1960 Abraham Robinson found that infinitesimals also provide a rigorous basis for the calculus. Many calculus courses still begin with the difficult limit concept, but others now adopt the easier to understand infinitesimal concept.
In differential calculus, in both the limit and infinitesimal versions, the velocity of a particle is represented by ds/dt, where s is the position of the particle and t the time at which the velocity is measured. In infinitesimal calculus, ds and dt are simply very small quantities.
Assume that s = t2, therefore at time t + dt, s + ds = (t + dt)2. From these equations we can determine ds in terms of dt using simple algebra, and moderate rigour, as follows:
ds = (t + dt)2 – t2 = ( t2 + 2tdt + dt2) – t2 = 2tdt + dt2
and
v = ds/dt = 2t + dt
Because dt is infinitesimal you can just ignore it, so the instantanous velocity is v = 2t. You get the same result using the limit version, with greater rigour. The infinitesimal derivation can also be made more rigorous.
References
 B. Crowell, "Calculus" (2003)
 J. Keisler, "Elementary Calculus" (2000) University of Wisconsin
 K. Stroyan "Foundations of Infinitesimal Calculus" (1993)

