What is a Derivative

Review: Slope of a line

The formal definition of a derivative is based on calculating the slope of a line.
Remember: The typical method for finding the slope is slope. We use any two points on the line and substitute them into the formula, as shown in the picture below.

Picture of slope from graph

Slope of a Secant Line

picture of secant line If we draw a line connecting two points on a curve this line is called a secant line. We can find its slope using the method above, slope. We have two points on the line, so using the formula, gives us the slope of the secant line.

The slope of the secant line gives you the average rate of change of the curve between the two points.


What does all of this have to do with a derivative?

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As you can see in the applet below, as the distances approach zero, the secant line approaches the tangent.

Secant to Tangent
picture of tangent A derivative is basically the slope of a curve at a single point.

The derivative is also often referred to as the slope of the tangent line or the instantaneous rate of change.

tangent as secant