Infinitesimals, hyperreals, and the transfer principle (Robinson 1966)
Hyperreal Number Line
The hyperreals *ℝ extend ℝ with infinitesimals ε (0 < ε < r for all r > 0 in ℝ) and infinities ω (ω > n for all n ∈ ℕ).
Every hyperreal x has a unique standard part st(x): the unique real infinitely close to x. This gives a rigorous foundation to "dx" in calculus.
Derivative via Infinitesimals
f'(x) = st[(f(x+ε) - f(x)) / ε]
The transfer principle (Łoś's theorem): any first-order sentence true in ℝ is true in *ℝ, and vice versa. So standard calculus results transfer instantly to *ℝ — and infinitesimal proofs transfer back.