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In calculus, a differential is an infinitessimally small change in a variable.
Definition
For this section x will be used simply as an example, a differential can use any variable. A differential is a change in a variable much like the familliar Δx. The difference is that a differential () is infinitely small and thus does not have an actual value.
Uses
A derivative (of a single variable equation) is a ratio of two differentials, typically notated which is the Leibniz notation equivalent of in Newton's notation for differentiation. This is a consequence of the slope equation where the only difference is the replacement of Deltas with differentials to reflect the fact that the x and y values are at one point, not across two. For a more in-depth explanation see derivative.
Integrals also use differentials. In fact, single variable integrals require a differential at the end. This differential represents the thickness of the rectangles making up the Riemann integral.
Confusion
It is common for math students (especially first year calculus students) to confuse differential and derivative. Although they sound similar, the mathematic meanings are distinctly distinguishable.
Although in some cases certain algebraic functions such as cancellation in fractions are applicable to differentials, it is important not to carry this convenient property too far. Differentials are not numbers (or variables) and cannot always be treated as numbers. Differentials have the same unit as the variable they are associated with.
History
Differentials were essential to the development of calculus and were discovered in the same time frame. However, the math innovation that made differentials more apparent and visible was Leibniz notation.
See also
"Differential (calculus)"