Differential of a function

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${\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}$
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In calculus, the differential represents the principal part of the change in a function ${\displaystyle y=f(x)}$ with respect to changes in the independent variable. The differential ${\displaystyle dy}$ is defined by

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

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

The precise meaning of the variables ${\displaystyle dy}$ and ${\displaystyle dx}$ 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 ${\displaystyle dx}$ and ${\displaystyle dy}$ are considered to be very small (infinitesimal), and this interpretation is made rigorous in non-standard analysis.