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How is the derivative related to the slope of 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).

In analytic geometry and calculus, the derivative of a function at a specific point is defined as the slope of the tangent line to the graph of that function at that point.21

While the slope of a straight line is constant and calculated as the ratio of the "rise" over the "run" between any two points, curves do not have a constant slope. To determine the slope at a single point on a curve, a limiting process is required. By choosing a second point on the curve and calculating the average slope, one can observe how that value changes as the distance between the two points approaches zero. The derivative is the limit of this ratio as the interval approaches zero, representing the instantaneous rate of change of the function.21