- Topics At a Glance
- Derivative as a Limit of Slopes
- Slope of a Line Between Two Points
- Slope of a Line Between Two Points on a Function
- Slope at One Point?
- Estimating Derivatives Given the Formula
- Estimating Derivatives from Tables
- Finding Derivatives Using Formulas
**Derivatives as an Instantaneous Rate of Change**- Average Rate of Change
- Instantaneous Rate of Change
- Units, Words, and Notation
- Tangent Lines
- How Tangent Lines Look
- When Tangent Lines Don't Exist
- Tangent Lines and Derivatives
- Tangent Line Approximation
- Finding Tangent Lines
- Using Tangent Lines to Approximate Function Values
- Differentiability and Continuity
- The Derivative Function
- Graphs of
*f*(*x*) and*f*' (*x*) - Theorems
- Rolle's Theorem
- The Mean Value Theorem
- In the Real World
- I Like Abstract Stuff; Why Should I Care?
- How to Solve a Math Problem
- Appendix: Speed v. Velocity

Approach the derivative from a different and more real-lifey direction.

There are two ways to think about speed while driving. The first way is to look at the speedometer, which shows how fast the vehicle going at that moment. The other way is to take the number of miles driven and divide by how long the drive took. The first way tells the "instantaneous" speedâ€”the speed at that instant. The second way gives average speed over the whole trip.

Although looking at the speedometer may seem like the best way to figure out speed, in order to relate this driving stuff properly to derivatives we need to talk about average speed first.