title: Increment theorem
text: # Increment theorem
  
  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
  
  $\Delta y=f'(x)\,\Delta x+\varepsilon \,\Delta x$
  
  for some infinitesimal ε, where
  
  $\Delta y=f(x+\Delta x)-f(x).$
  
  If Δx≠0{\textstyle \Delta x\neq 0} then we may write
  
  ${\frac {\Delta y}{\Delta x}}=f'(x)+\varepsilon ,$
  
  which implies that Δy/Δx $≈f^′(x){\textstyle$ {\frac {\Delta y}{\Delta x}}\approx f'(x)}, or in other words that Δy/Δx {\textstyle {\frac {\Delta y}{\Delta x}}} is infinitely close to $f^′(x){\textstyle$ f'(x)}, or $f^′(x){\textstyle$ f'(x)} is the standard part of Δy/Δx {\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
  
  $\Delta y=f'(x)\,\Delta x+\varepsilon \,\Delta x$
  
  holds with the same definition of Δy, but instead of ε being infinitesimal, we have
  
  $\lim _{\Delta x\to 0}\varepsilon =0$
  
  (treating x and f as given so that ε is a function of Δx alone).
  
  