While trying to teach myself calculus, I stumbled upon a BBC documentary called The Birth of Calculus. In the documentary, the narrator explains that Newton and other contemporary mathematicians were trying to find a universal method for finding tangents to polynomial curves.
Why was that a problem of interest to Newton and his contemporaries? What were they studying? Instantaneous velocities? Rates of change? Mechanical motion?
On
A tribute to Geometry was given by Newton himself in the preface of his book (for better reading see at archive.org)
"THE PRINCIPIA.
THE AUTHOR'S PREFACE
SINCE the ancients (as we are told by Pappus), made great account of the science of mechanics in the investigation of natural things : and the moderns, laying aside substantial forms and occult qualities, have endeavoured to subject the phaenomena of nature to the laws of mathematics, I have in this treatise cultivated mathematics so far as it regards philosophy. The ancients considered mechanics in a twofold respect ; as rational, which proceeds accurately by demonstration ; and practical. To practical mechanics all the manual arts belong, from which mechanics took its name. But as artificers do not work with perfect accuracy, it comes to pass that mechanics is so distinguished from geometry, that what is perfectly accurate is called geometrical , what is less so, is called mechanical. But the errors are not in the art, but in the artificers. He that works with less accuracy is an imperfect mechanic ; and if any could work with perfect accuracy, he would be the most perfect mechanic of all ; for the description of right lines and circles, upon which geometry is founded, belongs to mechanics. Geometry does not teach us to draw these lines, but requires them to be drawn ; for it requires that the learner should first be taught to describe these accurately, before he enters upon geometry ; then it shows how by these operations problems may be solved. To describe right lines and circles are problems, but not geometrical problems. The solution of these problems is required from mechanics ; and by geometry the use of them, when so solved, is shown ; and it is the glory of geometry that from those few principles, brought from without, it is able to produce so many things. Therefore geometry is founded in mechanical practice, and is nothing but that part of universal mechanics which accurately proposes and demonstrates the art of measuring. But since the manual arts are chiefly conversant in the moving of bodies, it comes to pass that geometry is commonly referred to their magnitudes, and mechanics to their motion. In this sense rational mechanics will be the science of motions resulting from any forces whatsoever, and of the forces required to produce any motions, accurately proposed and demonstrated. ..."
I will add that Newton hated polemics and public discussions : his geometric approach allowed him to make the 'entrance right' to any discussion high enough to avoid people unable to understand advanced mathematics...
Description of "the Reception of Newton's Principia". Concerning your question see page 5 for Newton and page 9 for Leibniz progress in the study of curves. Newton had learned Descartes (analytic) Geometry before learning Euclide's Geometry.
The famous astrophysicist Chandrasekhar undertook the translation of Newton's book in modern mathematics (amazon).
Finding slope of tangents and computing rate of change (instantaneous) are equivalent. See the wikipedia link : http://en.wikipedia.org/wiki/Derivative