I know particular mathematical topics did not arise in a vacuum, but around a host of other related ideas.
Here my functions are differentiable functions from and to euclidean spaces.
I know the "analytic" definition of the derivative of a function; the limit of an algebraic expression. For this we don't need to know about curves or geometry, so in principle we don't need to know anything about geometry to do calculus.
So I am wondering did the creators of calculus (Newton and Leibniz), and early users, always have in mind the "analytic geometry"? (if anyone has read their works). Did they think geometrically with slopes of tangent-lines as they came up with the idea of the derivative, and used it?