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In nonstandard analysis, a field of mathematics, the increment theorem states the following: Suppose a function y = f(x) is differentiable at x and that Δx is infinitesimal. Then $${\displaystyle \Delta y=f'(x)\,\Delta x+\varepsilon \,\Delta x}$$ for some infinitesimal ε, where $${\displaystyle \Delta y=f(x+\Delta x)-f(x).}$$

If ${\textstyle \Delta x\neq 0}$ then we may write $${\displaystyle {\frac {\Delta y}{\Delta x}}=f'(x)+\varepsilon ,}$$ which implies that ${\textstyle {\frac {\Delta y}{\Delta x}}\approx f'(x)}$, or in other words that ${\textstyle {\frac {\Delta y}{\Delta x}}}$ is infinitely close to ${\textstyle f'(x)}$, or ${\textstyle f'(x)}$ is the standard part of ${\textstyle {\frac {\Delta y}{\Delta x}}}$.

A similar theorem exists in standard Calculus. Again assume that y = f(x) is differentiable, but now let Δx be a nonzero standard real number. Then the same equation $${\displaystyle \Delta y=f'(x)\,\Delta x+\varepsilon \,\Delta x}$$ holds with the same definition of Δy, but instead of ε being infinitesimal, we have $${\displaystyle \lim _{\Delta x\to 0}\varepsilon =0}$$ (treating x and f as given so that ε is a function of Δx alone).

See also

• Nonstandard calculus
• Elementary Calculus: An Infinitesimal Approach
• Abraham Robinson
• Taylor's theorem

References

• Howard Jerome Keisler: Elementary Calculus: An Infinitesimal Approach. First edition 1976; 2nd edition 1986. This book is now out of print. The publisher has reverted the copyright to the author, who has made available the 2nd edition in .pdf format available for downloading at http://www.math.wisc.edu/~keisler/calc.html
• Robinson, Abraham (1996). Non-standard analysis (Revised ed.). Princeton University Press. ISBN 0-691-04490-2.

v t e Infinitesimals
History	Adequality Leibniz's notation Integral symbol Criticism of nonstandard analysis The Analyst The Method of Mechanical Theorems Cavalieri's principle
Related branches	Nonstandard analysis Nonstandard calculus Internal set theory Synthetic differential geometry Smooth infinitesimal analysis Constructive nonstandard analysis Infinitesimal strain theory (physics)
Formalizations	Differentials Hyperreal numbers Dual numbers Surreal numbers
Individual concepts	Standard part function Transfer principle Hyperinteger Increment theorem Monad Internal set Levi-Civita field Hyperfinite set Law of continuity Overspill Microcontinuity Transcendental law of homogeneity
Mathematicians	Gottfried Wilhelm Leibniz Abraham Robinson Pierre de Fermat Augustin-Louis Cauchy Leonhard Euler
Textbooks	Analyse des Infiniment Petits Elementary Calculus Cours d'Analyse

• Theorems in calculus
• Nonstandard analysis