Who invented differential calculus for rational functions?

[Vinay080]

Euler mentions in his preface of the book "Foundations of Differential Calculus" (Translated version of Blanton):

...Even now there is more that remains obscure than what we see clearly. As differential calculus is extended to all kinds of functions, no matter how they are produced, it is not immediately known that method is to be used to the vanishing increments of absolutely all kinds of functions. Gradually this discovery has progressed to more and more complicated functions, the ultimate ratio that the vanishing increments attain could be assigned long before the time of Newton and Leibniz, so that the differential calculus applied to only these rational functions must be held to have been invented long before that time. However, there is no doubt that Newton must be given credit for that part of differential calculus concerned with irrational functions...

I don't understand here, who/who all had invented/discovered the study-of-ultimate ratio (differential calculus) for rational functions long before (Newton and Leibniz), without knowing application of method to the vanishing icrements; if it was already invented, how does that differ from that of the study-of-ultimate ratio (differentail calculus) for irrational functions; it must be the same (??) method.

 

[HallsofIvy]

It was not "long" before Newton and Leibniz but Fermat developed a method he called "ad-equality" to find the tangent line to a graph at a point. Basically, it uses the idea of solving for the points at which y= ax+ b crosses the graph, the given point and one other, then determines the value of "a" that causes the two points to be the same.