Abstract
Which came first, the total differential or the partial derivative?
This seems like a simple question, because the total differential of a function $z=f(x,y)$ is defined in all the textbooks as
$$\begin{equation} dz = f_x(x,y) \, dx + f_y(x,y) \, dy, \cssId{totdiffdef}{\tag{1}} \end{equation}$$
with the obvious modifications when $f$ is a function of three or more variables.
If we understand the question “Which came first?” in the historical sense, however, we get the opposite answer, because the total differential is as old as the calculus itself, whereas partial derivatives were only defined in the 18th century.
In the integral calculus, we learn how to find integrals of functions. However, the first part of calculus, in which we learn how to find derivatives, is called the differential calculus, not the derivative calculus. Similarly, the course where we learn to solve equations involving derivatives is called Differential Equations.
What’s a differential, then? Modern textbooks usually define it in the chapter on applications of the derivative as $dy = f'(x) \, dx$, analogously to 1. The definition is often motivated as giving a local linear approximation to a function. This is particularly unsatisfying, given that the section on differentials frequently follows the section on Newton’s method for finding roots. We only appreciate the true value of these differentials in a later chapter, when we encounter integration by substitution.
In 1696, however, in the first calculus textbookFootnote1 BPS15, 2, the Marquis de l’Hôpital (1661–1704) defined the differential as follows:
Definition II. The infinitely small portion by which a variable quantity continually increases or decreases is called the Differential.
That is, a differential is an infinitely small increment in a variable. In the 17th century, some mathematicians were quite comfortable making arguments about infinitely small quantities, or infinitesimals, even though their reasoning was far from rigorous by current standards. Arguments involving infinitesimals date back at least as far as the determination of the area of a circle by Johannes Kepler (1571–1630). In fact, Archimedes (ca. 287–ca. 212 BCE) used the closely related method of indivisibles as a method of discovery (as opposed to proof) in his treatise The MethodFootnote2 Kat09, 103–110, 514–516.
A positive quantity was considered infinitely small if it was not zero, but smaller than any given positive quantity. It’s a fairly straightforward exercise in undergraduate analysis to show that no real number has this property. However, 17th-century mathematicians considered quantities in an informal sense, without precise definitions, as though numbers were given a priori. Infinitesimals could be conceived as being related to finite quantities in analogy to how finite quantities are related to infinity—roughly speaking, if $\alpha$ is infinitesimal, then $\alpha : 1\Colon 1 : \infty$. For example, even though $\infty$ wasn’t rigorously defined, it was clear to them that $\infty + 1 = \infty$. So it must be that $1 + \mathrm{infinitesimal} = 1$. More generally, if $x$ is a finite quantity (a real number) and $dx$ is an infinitesimal increment in $x$, then $x + dx = x$, even though $dx \ne 0$. This strange state of affairs was turned into an axiom by Johann Bernoulli (1667–1748); see Postulate I in Section 2.2.