How did Guillaume de l'Hôpital "devise" his rule?

Question

I saw on Wikipedia, the proof of general case of L'Hopital's rule was given by "Taylor, 1952".
But L'Hopital was born in 1661, then how he came to know about this "rule", and if he just conjectured the rule, then why was it called "rule"?
And also, at that time, calculus was just born, so how did he "devise" the rule, is there any proof of his own? If yes, please show.
Edit: One of the comments suggests that the rule was discovered by John Bernoulli. But then why L'Hopital was accused of plagiarism? And why is the rule named after L'Hopital? And still the question remains the same: "But Bernoulli was born in 1667, then how he came to know about this "rule", and if he just conjectured the rule, then why was it called "rule"?
And also, at that time, calculus was just born, so how did he "devise" the rule, is there any proof of his own? If yes, please show."

Answer 1

See Analyse des Infiniment Petits pour l'Intelligence des Lignes Courbes (1696, and several following editions) :

is the first textbook published on the infinitesimal calculus of Leibniz. It was written by the French mathematician Guillaume de l'Hôpital, and treated only the subject of differential calculus.

See C.Truesdell, The New Bernoulli Edition Isis, Vol. 49, No. 1. (Mar,1958), pp.54–62, discussing the strange agreement between Bernoulli and l'Hôpital.

You can try to browse the book to "taste" how different is the calculus from the modern presentation... but it works, see Example II, page 17 :

Soit $x=\frac {ay} b$, dont la différence est $dx=\frac {ady} b$.

See :

• Margaret Baron, The Origins of Infinitesimal Calculus (1969 - Dover edition)

or :

• Carl Boyer, The History of the Calculus and Its Conceptual Development (1959 - Dover edition); see page 238-on.

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On "l'Hospital's rule", see :

• C.H.Edwards Jr, The historical development of the calculus (1979), page 269 for a discussion :

L'Hospital's argument, which is stated verbally without functional notation [...] amounts simply to the assertion that

$$\frac {f(a + dx)}{g(a + dx)} = \frac {f(a)+f'(a)dx}{g(a) + g'(a)dx} = \frac {f'(a)dx}{g'(a)dx}=\frac {f'(a)}{g'(a)}$$

provided that $f(a) = g(a) = 0$. He concludes that, if the ordinate $y$ of a given curve "is expressed by a fraction, the numerator and denominator of which do each of them become $0$ when $x = a$" then "if the differential of the numerator be found, and that is divided by the differential of the denominator, after having made $x = a$, we shall have the value of [the ordinatey when $x = a$]".

In the French text, page 145 of the 2nd ed (1716) :

Proposition I. Problême. Soit une ligne courbe AMD (AP = x, PM = y, AB = a - Fig.130 ) telle que la valeur de l’appliquée y soit exprimée par une fraction, dont le numérateur & le dénominateur deviennent chacun zero lorsque x = a, c’est à dire lorsque le point P tombe sur le point donné B. On demande quelle doit être alors la valeur de l’appliquée BD. [Solution: ]...si l’on prend la difference du numérateur, & qu’on la divise par la difference du enominateur, apres avoir fait x = a = Ab ou AB, l’on aura la valeur cherchée de l’appliquée bd ou BD."

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How he "devised" it ? Who knows ... But the "power" of Leibnizian symbolism was its capability to apply algebraic "transformation rules" to infinitesimals, in spite of all "foundational" issues.