Secant Lines, Tangent Lines, and Limit Definition of a Derivative

(Note: this page is just a brief review of the ideas covered in Group. It is meant to serve as a summary only.)

A **secant line** is a straight line joining two points on a function. (See below.) It is also equivalent to the **average rate of change**, or simply the **slope** between two points.

The **average rate of change **of a function between two points and the **slope** between two points are the same thing.

Secant line = Average Rate of Change = Slope |

A **tangent line** is a straight line that touches a function at only one point. (See above.) The tangent line represents the **instantaneous rate of change** of the function at that one point. The slope of the **tangent line** at a point on the function is equal to the **derivative** of the function at the same point (See below.)

Tangent Line = Instantaneous Rate of Change = Derivative |

Let's see what happens as the two points used for the secant line get closer to one another. Let Dx represent the distant between the two points along the x-axis and determine the limit as Dx approaches zero.

As the two points used for the secant line get closer to one another, the average rate of change becomes the instantaneous rate of change and the secant line becomes the tangent line. (See animation above.)