Who invented differential calculus for rational functions?

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1. Oct 6, 2015

Vinay080

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

Last edited: Oct 6, 2015
2. Oct 7, 2015

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.