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I need help ascertaining which calculus notation is more preferred amongst mathematicians according to their home countries, and why. I have tried looking for a census or survey of some sort that answers my question. Does anyone have access to this information?

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It is often more useful in some contexts. I will give an example:

  1. Given some function $f(x)$, we can either write $\dot{f}$ (Newton) or $\frac{df}{dx}$ (Leibniz). It just boils down to preference. However given some function $f(x, y)$ if we take the implicit derivative with respect to only one variable we typically don't write $\dot{f}(x, y)$ as that doesn't contain any additional information on which variable $f$ is being differentiated with respect to. $$\frac{d}{d x}f(x, y)$$ Makes 'more sense' to use. An example is the use of Leibniz's notation in multivariable calculus

Another reason

  1. Given some function $y$, we can use the chain rule: $$\frac{dy}{dx} = \frac{dy}{dz}\frac{dz}{dx}$$ Which works and suggests that Leibniz's notation treats derivatives as fractions. This is due to the fact that Leibniz thought of derivatives as quotients, and while this is mathematically incorrect, it stuck around for reasons like the example above

Different countries and/or institutions will have different historical influences and curriculum choices that could determine whether one notation is preferred over the other. However, it is important to note that both notations are widely used and accepted. In modern day mathematical literature it seems preferrential to use Leibniz's notation for the reasons above

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    $\begingroup$ About 1. The modern notation for partial derivatives is due to Adrien-Marie Legendre, not Leibniz. $\endgroup$
    – Kurt G.
    Dec 13, 2023 at 13:56
  • $\begingroup$ @KurtG. I should have used regular $dx$ notation previously stating it is implicit differentiating- Thanks for pointing that out $\endgroup$
    – Mako
    Dec 13, 2023 at 13:59
  • $\begingroup$ If $f:\mathbb R^2\to\mathbb R$ is a differentiable function, then I would interpret $f'(x,y)$ to mean the Fréchet derivative of $f$ at $(x,y)$. In this context, there is no such thing as "the derivative with respect to a variable". In order to make sense of the notation $\frac{d}{dx}f(x, y)$, you have to be thinking of $f(x,y)$ as a function of one variable $x$, so that $y$ is actually a notational shorthand for $g(x)$, where $g$ is some function of one real variable. $\endgroup$
    – Joe
    Dec 13, 2023 at 14:06
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    $\begingroup$ Another remark: the notation $f'(x)$ is due to Lagrange. Newton wrote $\dot f\,.$ $\endgroup$
    – Kurt G.
    Dec 13, 2023 at 14:10
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    $\begingroup$ Apparently Newton's notation was looked down upon by continental mathematicians of that time. They called it "dot-age". The English word "dotage" means senility. Rough times, even then. $\endgroup$
    – Kurt G.
    Dec 13, 2023 at 14:21
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My favorite notation is Weierstraß's notation $$ f(x_0+v)=f(x_0)+J_{x_0}\cdot v + r(v) $$ where for the remainder function holds $\displaystyle{\lim_{v \to 0}\dfrac{r(v)}{\|v\|v}=0}.$ I like it since it clearly distinguishes the different parts of a derivative: function $f$, location $x_0$, direction $v$, linear approximation $\dfrac{f(x_0+h)-f(x_0)}{h}$, linear functional $v\mapsto J_{x_0}\cdot v$ and higher order terms $r(v).$ This notation where $J$ is the Jacobi matrix and $v$ the direction of change becomes $$ f(x_0+h)=f(x)+\underbrace{\left. \dfrac{d}{dx}\right|_{x_0}f(x)}_{=f'(x_0)} \cdot h + O(h^2) $$ in the case of only one real dimension of $x$. All information is contained:

  • a derivative is a local property (valid in a neighborhood of $x_0$)
  • a derivative is a slope ($f'(x_0)$)
  • the derivative as the result of differentiation is again a function ($x_0\mapsto f'(x_0)$)
  • the derivative is a linear function with the direction as its variable ($h \mapsto f'(x_0)\cdot h$))
  • differentiation is a linear operator ($D=\frac{d}{dx}:f\mapsto f'$)
  • and a derivative is always directional ($h$)

We get very different perspectives depending on which of these quantities we consider a variable, and which we consider a parameter and misunderstandings are often based on confusion about which perspective is meant in a specific context. And in more than one dimension, even more perspectives are possible (linear functionals, sections of tangent bundles, etc.). Physicists use all of them.

So my answer to your question is: use the notation that fits best to your perspective. Leibniz's notation strengthens the perspective of a linear operator but only because people usually forget the locality of differentiation when they write $$ \dfrac{df}{dx}\;\text{ and mean }\;\left. \dfrac{d}{dx}\right|_{x=x_0}f(x) $$ whereas Newton's notation emphasizes on the resulting function $$ x_0\longmapsto f'(x_0) $$ as the result of differentiation which lacks the notation of a direction since there is only one direction in the one-dimensional case.

Summary: Newton's notation is harder to generalize than Leibniz's notation, and both are still less informative than Weierstraß's notation. To decide Leibniz or Newton means to decide what you want to neglect.

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