- 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

The **derivative** of the function *f* at *x* = *a* is the slope of the function *f* at *x* = *a*.

What the wha—? We typically find the slope of a line between two points. But what does it mean to find the "slope" of a curvy function that isn't a straight line? And how do we find the "slope" of something if we're only given one point?

These questions are actually answerable, thanks to calculus. We can also thank Fig—ahem, *Isaac* Newton for figuring this stuff out. Snack break, anyone?