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The standard part function is used to define the derivative of a function ''f''. If ''f'' is a real function, and ''h'' is infinitesimal, and if ''f''′(''x'') exists, then
:<math>f'(x) = \operatorname{st}\left(\frac {f(x+h)-f(x)}h\right).</math>
Alternatively, if <math>y=f(x)</math>, one takes an infinitesimal increment <math>\Delta x</math>, and computes the corresponding <math>\Delta y=f(x+\Delta x)-f(x)</math>. One forms the ratio <math>\frac{\Delta y}{\Delta x}</math>. The derivative is then defined as the standard part of the ratio:
:<math>\frac{dy}{dx}=\text{st}\left( \frac{\Delta y}{\Delta x} \right)</math>.
==See also==
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