# Differential of a function

In calculus, the **differential** represents the principal part of the change in a function *y* = *f*(*x*) with respect to changes in the independent variable. The differential *dy* is defined by

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where is the derivative of *f* with respect to *x*, and *dx* is an additional real variable (so that *dy* is a function of *x* and *dx*). The notation is such that the equation

holds, where the derivative is represented in the Leibniz notation *dy*/*dx*, and this is consistent with regarding the derivative as the quotient of the differentials. One also writes

The precise meaning of the variables *dy* and *dx* depends on the context of the application and the required level of mathematical rigor. The domain of these variables may take on a particular geometrical significance if the differential is regarded as a particular differential form, or analytical significance if the differential is regarded as a linear approximation to the increment of a function. Traditionally, the variables *dx* and *dy* are considered to be very small (infinitesimal), and this interpretation is made rigorous in non-standard analysis.