Learning Outcomes

- Use the definition of derivative to find the derivative function of a given function.
- State the connection between differentiability and continuity.
- Describe three conditions for when a function does not have a derivative.
- Explain the meaning of a higher-order derivative

**Textbook**

- Chapter 3.2 The Derivative as a Function

**Textbook Assignment**

- p. 243: 54, 55, 57, 58, 59, 61, 62

**WeBWorK** **Assginment**

- Derivatives-Functions

**Exit problems ****of the session **

- Let . Use the definition of a derivative to find .
- Let . Use the definition of a derivative to find .

** Key Concepts**

- The derivative of a function is the function whose value at is :

- If a function is differentiable at a point, then it is continuous at that point. A function is
at a point if it is**not differentiable**, if it has a*not continuous at the point*at the point, or if the*vertical tangent line*.*graph has a sharp corner or cusp* - Higher-order derivatives are derivatives of derivatives, from the second derivative to the derivative.

#### Videos and Practice Problems of Selected Topics

**The definition of the derivative as a limit**

- Formal definition of the derivative as a limit (15:42) The derivative of function at is the limit of the slope of the secant line from to as approaches .
- The derivative of at any point using the formal definition (11:04)
- * Practice: Finding tangent lines using the formal definition of a limit. (6 questions with a guiding text)
- Worked example: derivative as a limit (5:45) Find when .
- Worked example: derivative from limit expression (5:16) Interpreting a limit expression as the derivative of at the point .
- * Practice: Derivative as a limit. (4 problems)
- * Derivative notation: a review of three notations. (two problems with a guiding text)

**Connecting differentiability and continuity**

- Differentiability and Continuity (9:37) Defining differentiability and getting an intuition for the relationship between differentiability and continuity.
- Differentiability at a point: graphical (5:38) Find the points on the graph of a function where the function isn’t differentiable.
- * Practice: Differentiability at a point: graphical. (4 problems)
- Differentiability at a point: algebraic (5:01) Is the function continuous/differentiable at ?
*(Optional)* - Differentiability at a point: algebraic (6:21) Is the function continuous/differentiable at ?
*(Optional)* - * Practice: Differentiability at a point: algebraic. (4 problems)
*(Optional)*