However, they were unable to provide a rigorous foundation for the existence of these quantities, and mathematicians instead began to embrace the now-ubiquitous epsilon-delta approach to the foundations of calculus, avoiding the notion of infinitesimal numbers altogether. 201.Newton and Gottfried Leibniz both used infinitesimals-numbers which are nonzero, yet smaller in magnitude than any real number-in the early 1700s to describe instantaneous rates of change in their developments of calculus. There is a discussion of this in Boyer, at p. He had a notation $o$ for an infinitesimal change in the independent variable, so that if $x$ depends on $t$, then what in Leibniz notation would be written as $dx$ would be notated in Newton's notation as $\dot$, omitting the $o$ when context made it clear that what was intended was an infinitesimal change in $x$. Newton's notation was unclearly presented, and many people didn't understand how he intended it to be used. It works whether you want to think in terms of variables or functions, limits or infinitesimals. Unlike Newton's notation, it makes it easy to do dimensional analysis, and it works well when you have lots of different variables that you might be differentiating or integrating with respect to. Leibniz's notation has some objective advantages.
For example, in physics, if you have a function of position and time, it's common to use dots for time derivatives and primes for spatial derivatives. For example, see Hutton, 1807,, which uses the dot notation and terms like "fluent." We still do use elements of Newton's notation in many fields. Many textbooks in English did use Newton's notation and terminology for a long time. Moreover, they generalise to the infinite-dimensional context seamlessly, unlike the usual calculus for which there are many differing techniques). (It's worth adding that doing differential geometry with intuitionistic logic allows the introduction of infinitesimals that is much closer to how Newton envisaged them, his fluxions, rather than the traditional epsilon-delta techniques of traditional analysis. In other words, the notation associated with their names reflected their interests. Had Newton been more interested in geometry than physics, and Liebniz more interested in physics than geometry it would have been likely we would have seen their notations being swapped. Thus Liebniz's notation is more natural here. This should indicate the degree, the dependent and independent variables. When one is interested in the calculus for its own sake, then a more comprehensive notation is necessary. Buy since we are only interested in the first two, 1 & 2, we don't have to indicate the degree by a numerical prefix (as they do in some notations), we can simply indicate it by a single or double dot. Hence he needed only to indicate the degree of the derivative.
Since the dependent variable implicitly understood, there is no need for the notation to reflect this. As a physicist he was mostly interested in the first and second derivatives of time. This is also used for the Frechet and Gateaux derivative (which is implicitly used in the notation for tangent bundles in differential geometry).
For example, there is also Heaviside's operational D. There are many notations for derivatives as the concept has been expanded in many different ways.
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