# Who was the first to understand a derivative and integral as both giving rise to new functions?

NOTE: Even if not called "functions" surely someone understood that returning something other than a numerical value was significant?

If I understand it correctly, Newton, for example, was trying to get a single value at either a point in the case of the derivative or under a curve between two points. This is different from saying, the integral of a function is another function (similar with a derivative).

Am I wrong and Newton already understood that both operations produce new functions?

• My guess is that a distinction like this would be obvious (even banal or trite) to anyone who cared to think about it then, and it only became explicit (and still obvious) when mathematical formalism rose to where one cared about making such distinctions. Probably this question is too steeped in our present viewpoints and emphasis for an answer to have any significant historical merit. That said, some of the letters from/to Darboux reproduced here might be of interest to you. (continued) Commented May 2 at 12:30
• Probably around pp. 75-78; I don't know where my decades-old photocopy is, in order to give a more specific reference, and the online paper is behind a paywall that I don't have access to. The letters I'm thinking of involve Darboux's attempt to explain to someone why a certain piecewise-defined function is a counterexample to something (the other person holding the view that using different formulas is not valid, or something like that). Commented May 2 at 12:30
• The concept of "function" did not take its modern form until 19th century, so this conception evolved gradually without any "first". Oresme was producing graphs of velocity from graphs of position already in 14th century. Newton expanded fluxions into power series, so they were "functions". Leibniz and Bernoulli had the idea of functional dependence by the end of 17th century, and applied it to derivatives and integrals in particular. Euler codified these ideas in his mid-18th century textbooks, see MacTutor, The function concept. Commented May 2 at 23:03
• See here for some details about the history of the "fucntion" concept and see also Leibniz. Commented May 3 at 6:47

While logarithms were introduced by Napier already at the beginning of the 17th century, the natural logarithm was introduced in 1649 by a little-known student of Gregoire de Saint-Vincent, precisely by means of the quadrature of the hyperbola. See this for details.

• Are you saying that it was understood by Saint-Vincent that an equation could have a result that was not a quantity but in fact another equation and that it was essentially stated this way? I imagine this was a big step forward in mathematical thinking. Commented May 2 at 18:11
• That's what the wiki article suggests. They thought of it in terms of a variable parameter $t$. Commented May 3 at 10:21

If you're looking for an early quote where someone uses the expression 'function' and familiar notation, then the earliest person to look into is Leibniz (because he coined those terms), and it's likely that you can find it in his or Bernoullis works. In the meantime, here are some quotes from Euler's book on differential calculus (1748):

Paragraph 120:

Quoniam igitur vidimus differentiale primum ipfius $$y$$, fi $$y$$ fuerit functio quaecunque ipfius $$x$$, habiturum effe huiusmodi formam $$P \cdot \omega$$; ob $$\omega=d x$$, erit $$d y=P d x$$. Qualiscunque fcilicet fuerit $$y$$ functio ipfius $$x$$, eius differentiale $$d y$$ exprimetur certa quadarn functione ipfius $$x$$, pro qua hic ponimus $$P$$, per differentiale ipfius $$x$$, nempe per $$d x$$ multiplicata.

My translation and emphasis:

Since we have seen that the differential of $$y$$, when $$y$$ is any function of $$x$$, takes the form $$P\cdot \omega$$ and since $$\omega=dx$$, we obtain $$dy=Pdx$$. So, whatever function $$y$$ is of $$x$$, its differential $$dy$$ is always the product of a function of $$x$$, which we have here denoted $$P$$, with the differential of $$x$$, hence $$dx$$.

Here $$P$$ is of course the derivative, later denoted $$\frac{dy}{dx}$$. More explicitly he writes in paragraph 175:

[...] Hancobrem fi $$y$$ fuerit functio algebraica ipfius $$x$$, erit quoque $$\frac{d y}{d x}$$ functio algebraica ipfius $$x$$.

My translation:

Hence if $$y$$ is an algebraic function of $$x$$, also $$\frac{d y}{d x}$$ must be an algebraic function of $$x$$.