Introduction

We have discussed a secant line to the graph of a function from to , and the fact that the slope, , of this line is the average rate of change of the function on the interval , . Furthermore, recall that we can let , so that the slope expression becomes where is the horizontal distance between the points.

One of the main objectives of Calculus is to understand instantaneous rates of change, as opposed to average rates of change. Namely, what is the behavior of the expression when gets very small? Geometrically, making become very small is making the secant line through and approach a certain line — the tangent line to the graph of at the point . This is demonstrated in the figure below, where the secant line is blue, and the line tangent to the graph at is in red.

Follow the link below to an example Desmos graph where you can see the effect of changing the value of on the secant line in real-time.

The slope of such a tangent line, when it exists, is called the derivative of at . Here, we’ll discuss difference quotients and several examples, to prepare you to learn those things in more detail in a future Calculus class.

Definitions and examples