Britannica Sources used in this answer

How can a derivative be found from the slope of a chord as the chord becomes shorter?

Created from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles.

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 slope of a chord (or secant line) connecting two points on a curve represents the average rate of change of a function over an interval.

To find the derivative from this chord, you consider the interval between two points, typically denoted as $t$ and $t + h$ . The slope of the chord is calculated as the change in the function's value divided by the change in the variable ( $h$ ). As the distance $h$ becomes smaller and smaller, the two points on the curve move closer together. If this slope approaches a specific limiting value as $h$ tends to zero, that limit is defined as the derivative of the function, denoted as $f^{'} (x)$ .12

Geometrically, as $h$ approaches zero, the direction of the chord stabilizes, and the chord increasingly approximates the tangent line to the graph at that point. Consequently, the derivative represents the instantaneous rate of change of the function at that specific point.13