I'm asking who was the first to use and define velocity and acceleration in the modern, now standard way, with velocity being the first derivative of position and acceleration being the second derivative of the position vector. I have heard that Galileo Galilei, but he was born before the invention of calculus. I am also aware that vector analysis as we know it did not start until the 19th century.
2$\begingroup$ Yes, Galileo is definitely modern, but the definitions are in terms of ratio and not of "calculus". $\endgroup$– Mauro ALLEGRANZAMay 7, 2021 at 5:51
$\begingroup$ Basically, Newton invented the calculus exactly to express "rate of change"; the fluxion is the instantaneous rate of change of a changing quantity (a fluent), i.e. a derivative with respect to time. $\endgroup$– Mauro ALLEGRANZAMay 7, 2021 at 5:52
$\begingroup$ See the post Galileo's Discussion of Uniform Motion. $\endgroup$– Mauro ALLEGRANZAMay 7, 2021 at 7:46