- 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
Introduction to :
The "derivative of f at a," written f ' (a), is a number that is equal to the slope of the function f at a.
For any differentiable function f there is another function, known as the derivative of f and written f ' (x). We write f '(x) to show that this is a function.
We can calculate the derivative function using the limit definition in the same way we calculated the value of the derivative at a point using the limit definition.
Practice:
Let f(x) = x2. Calculate f'(x). |
Let f be a line. That is, f(x) = mx + b where m and b are constants. Show that f'(x) = m. |
Let f(x) = x3. Find f'(x). |
Let f(x) = x2. We found that f'(x) = 2x. Find f(3). |
Let f(x) = x3. Use the formula we found for f'(x) to evaluate
- f'(4)
Let f(x) = x3. Use the formula we found for f'(x) to evaluate
- f'(-5)
Let f(x) = x3. Use the formula we found for f'(x) to evaluate
