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How did Fermat use algebra to find a tangent line to a curve?

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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).

Pierre de Fermat revolutionized analytic geometry by demonstrating that the properties of curves, including the determination of tangent lines, could be solved systematically using algebra.12

In the mid-1630s, Fermat developed an algorithm for finding tangents that was adapted from a procedure he had created to identify the maximum and minimum points of a curve. His method relied on finite algebraic ideas rather than the formal infinitesimal calculus later developed by Isaac Newton and Gottfried Leibniz. By expressing a curve as a polynomial equation, Fermat could apply algebraic manipulations to determine the slope at any point.321

While Fermat’s original account of this method was brief and lacked a detailed mathematical justification, it was functionally equivalent to differentiation. His work allowed for the routine analysis of a vast class of algebraic curves, moving beyond the isolated, difficult geometric cases that had challenged mathematicians since antiquity.321