In calculus, the differential represents the principal part of the change in a function with respect to changes in the independent variable. The differential is defined by
where is the derivative of f with respect to , and is an additional real variable (so that is a function of and ). The notation is such that the equation
holds, where the derivative is represented in the Leibniz notation , and this is consistent with regarding the derivative as the quotient of the differentials. One also writes
The precise meaning of the variables and 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 and are considered to be very small (infinitesimal), and this interpretation is made rigorous in non-standard analysis.
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In calculus, the differential represents the principal part of the change in afunction y = f ( x ) {\displaystyle y=f(x)} with respect to changes in the...
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