The Marquis de l’Hôpital, a French nobleman living by private means, is known for the first printed book on the newcalculus. He served briefly as a cavalry officer, but resigned because of his extreme nearsightedness to devote his energies entirely to mathematics. In his time the recently invented calculus was fully understood only by Newton, Leibniz and the Bernoulli brothers. In 1691-92, when John Bernoulli spent over half a year in Paris, he was generously compensated for giving the young Marquis private lessons on this powerful new method. In return for a monthly allowance, Bernoulli was induced to continue the instruction by letter; the agreement was that he would communicate his future mathematical discoveries exclusively to l’Hôpital to be used as the Marquis saw fit. L’Hôpital eventually felt that he understood the material well enough to compose a proper textbook on it.

L’Hôpital’s Analyse des infiniment petits, published in 1696, contains an account of the differential calculus as conceived by Leibniz and learned from Bernoulli. In its preface l’Hôpital freely acknowledges his debt to the two mathematicians, saying, "I have made free use of their discoveries." The successive reprintings of the Analyse (1716, 1720 and 1768) made the calculus known throughout Europe. In 1730 it was translated into English, supplemented by the translator with work on the integral calculus; in tribute to Newton, the book’s derivative notation was changed to the fluxional "dottage" of their English hero. L’Hôpital is nowadays remembered in the name of his "0/0 rule," a rule for finding the limiting value of a quotient whose numerator and denominator both tend to zero. His statement of the rule is not entirely in accord with modern use. Making use of limit notation, which was unavailable to l’Hôpital, a reasonable rendition of his statement would be:

The Analyse dominated the field for the next 50 years, finally to find a worthy rival in Euler’s great treatises of the 1750’s.