- 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

In this section we need to find derivatives "analytically," also known as "using the limit definition."

**Be Careful:** "Find the derivative using the limit definition" does not mean estimating the derivative like we did earlier . There, we were estimating. Now, we'll be exact.

Let *f*(*x*) = *x*^{2} + 1. Find *f'*(*1*) using the limit definition of the derivative.We have *a *= 1 and *f*(*x*) = *x*^{2} + 1, so we find

Be careful here. Make sure to plug in (1 + *h*) to *f*, and don't forget about the extra + 1 at the end of the function *f*:

We conclude *f'*(1) = 2.

Let *f*(*x*) = *x*^{2} - *x*. Use the limit definition of the derivative to find *f'*(*2*).

We have *a* = 2.

Here's another place to be careful. It's possible to lose *x*'s and *h*'s in confusing spots like this, evaluate *f*(2 + *h*) and *f*(*2*) separately.

Putting things back in,

We conclude that *f*'(2) = 3.

If derivatives take this long, how does anyone finish their homework in time to watch *The Late Show*? Why doesn't anyone's arm ever fall off during a calc exam? In answer to these questions, yes, there are easier ways of calculating derivatives. However, "the limit definition of the derivative" is important, and it will be on the test.

Example 1

Let |

Example 2

Use the limit definition of the derivative to find |

Example 3

Use the limit definition of the derivative to find |

Exercise 1

Using the limit definition, find *f'*(0) if *f*(*x*) = *x*^{2}.

Exercise 2

Using the limit definition, find *f*'(-1) if *f*(*x*) = 1 - *x*^{2}.

Exercise 3

Using the limit definition, let *f*(*x*) = *x*^{3}. Find *f'*(1).

Exercise 4

If it exists, use the limit definition to find the derivative of *f*'(-1) if *f*(*x*) = *x*^{2}-* x.
*

Exercise 5

Use the limit definition to find the derivative, if it exists: *f'*(*1*) where .

Exercise 6

Use the limit definition to find the derivative, if it exists: *f'*(*0*) where *.*

Exercise 7

Use the limit definition to find the derivative, if it exists: *f'*(*0*) where *f*(*x*) = *x*^{1/3}.