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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}=\mathrm{st}\left( \frac{\Delta y}{\Delta x} \right)</math>.
Similarly, given a function <math>f</math> on <math>[a,b]</math>, one defines the integral <math>\int_a^b f(x)dx</math> as the standard part of an infinite Riemann sum <math>S(f,a,b,\Delta x)</math> when the value of <math>\Delta x</math> is taken to be infinitesimal, exploiting a [[hyperfinite set|hyperfinite]] partition of the interval [a,b].
==See also==
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