# When did derivative mean not only "slope of tangent" but also "instantaneous rate of change"?

When did derivative mean not only "slope of tangent" but also "instantaneous rate of change"?

Fermat was interested in minima and maxima, and realized these occur when the tangent has zero slope. When was it realized that this slope of tangent was the same thing as an instantaneous rate of change, i.e. velocity?

The post History of the derivative/tangent of a curve discusses the history of the tangent notion, but not the instantaneous rate of change notion.

• I'm pretty sure that Newton's definition of a fluxion (derivative) was as the rate of change of the fluent (the changing quantity). For Newton, fluxion was synonymous with velocity.
– nwr
Jun 12, 2023 at 16:57
• Background: Sherry, "On Instantaneous Velocity" (1986) jstor.org/stable/27743785, on the shift from Aristotle in the Middle Ages (in particular by Oresme) that introduced a "language-game" that permitted discussion and analysis of instantaneous velocity, in particular under uniform acceleration. Jun 12, 2023 at 17:44
• The relation between tangency and velocities/rates of change was known long before derivatives. Archimedes uses it implicitly in drawing a tangent to the spiral, Oresme and Oxford calculators made it explicit for simple graphs, Galileo likely picked it up from them. His student Torricelli used it systematically to find velocities of projectiles with power equations of motion via tangents to higher parabolas. By Newton's time, the correspondence was well-established, see Boyer, History of Calculus, ch. IV. Jun 12, 2023 at 19:32

As far as Fermat is concerned, it should be noted that he did not work with any notion of "slope" at all, as far as I am aware. He did realize that at minima and maxima of an expression (he didn't use functions, either), the change of the expression is only quadratic in the change $$E$$ in the variable.