Differential (infinitesimal)

The term differential is used in calculus to refer to an infinitesimal (infinitely small) change in some varying quantity. For example, if x is a variable, then a change in the value of x is often denoted Δx (pronounced delta x). The differential dx represents an infinitely small change in the variable x. The idea of an infinitely small or infinitely slow change is extremely useful intuitively, and there are a number of ways to make the notion mathematically precise.

Using calculus, it is possible to relate the infinitely small changes of various variables to each other mathematically using derivatives. If y is a function of x, then the differential dy of y is related to dx by the formula

where dy/dx denotes the derivative of y with respect to x. This formula summarizes the intuitive idea that the derivative of y with respect to x is the limit of the ratio of differences Δy/Δx as Δx becomes infinitesimal.

There are several approaches for making the notion of differentials mathematically precise.

These approaches are very different from each other, but they have in common the idea to be quantitative, i.e., to say not just that a differential is infinitely small, but how small it is.

Infinitesimal quantities played a significant role in the development of calculus. Archimedes used them, even though he didn't believe that arguments involving infinitesimals were rigorous.Isaac Newton referred to them as fluxions. However, it was Gottfried Leibniz who coined the term differentials for infinitesimal quantities and introduced the notation for them which is still used today.

In Leibniz's notation, if x is a variable quantity, then dx denotes an infinitesimal change in the variable x. Thus, if y is a function of x, then the derivative of y with respect to x is often denoted dy/dx, which would otherwise be denoted (in the notation of Newton or Lagrange) ẏ or y ′. The use of differentials in this form attracted much criticism, for instance in the famous pamphlet The Analyst by Bishop Berkeley. Nevertheless, the notation has remained popular because it suggests strongly the idea that the derivative of y at x is its instantaneous rate of change (the slope of the graph's tangent line), which may be obtained by taking the limit of the ratio Δy/Δx of the change in y over the change in x, as the change in x becomes arbitrarily small. Differentials are also compatible with dimensional analysis, where a differential such as dx has the same dimensions as the variable x.

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