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How did Newton and Gottfried Leibniz use coordinates in the development of calculus?

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Diagram showing a double cone intersected by planes at different angles, producing the conic sections: a horizontal plane creates a circle, a tilted plane creates an ellipse, a plane parallel to the cone's side forms a parabola, and a steep plane produces a hyperbola. Each section is labeled, with the axis and apex of the cone indicated. — analytic geometry The conic sections result from intersecting a plane with a double cone, as shown in the figure. There are three distinct families of conic sections: the ellipse (including the circle), the parabola (with one branch), and the hyperbola (with two branches).

Isaac Newton and Gottfried Leibniz independently revolutionized mathematics by using Cartesian algebra and coordinate systems to synthesize earlier geometric results into general algorithms. By applying these algebraic methods to curves defined by coordinate equations, they moved beyond the limitations of classical geometric intuition.12

Newton initially used coordinates to reformulate results using infinite series, expressing variables as "fluents" and their rates of change as "fluxions". While he later shifted toward a geometric theory of limits in his Principia, his early work relied on analytic processes to establish the inverse relationship between tangents and areas.1

Leibniz, conversely, focused on developing a powerful notation that expressed calculus in full generality. He treated coordinates as continuous quantities, using the symbols $d$ and $\int$ to represent infinitesimal differences and sums. His approach allowed for a uniform application of differentiation and integration across a wide class of functions, directly leading to the fundamental theorem of calculus.213