Learning Outcomes

1. Use the definition of derivative to find the derivative function of a given function.
2. State the connection between differentiability and continuity.
3. Describe three conditions for when a function does not have a derivative.
4. 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

1. Let .  Use the definition of a derivative to find 2. 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 not differentiable at a point if it is not continuous at the point, if it has a vertical tangent line at the point, or if the 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
1. 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 .
2. The derivative of at any point using the formal definition (11:04)
3. * Practice: Finding tangent lines using the formal definition of a limit. (6 questions with a guiding text)
4. Worked example: derivative as a limit (5:45) Find when .
5. Worked example: derivative from limit expression (5:16) Interpreting a limit expression as the derivative of at the point .
6. * Practice:  Derivative as a limit. (4 problems)
7. * Derivative notation: a review of three notations. (two problems with a guiding text)
• Connecting differentiability and continuity
1. Differentiability and Continuity (9:37) Defining differentiability and getting an intuition for the relationship between differentiability and continuity.
2. Differentiability at a point: graphical (5:38) Find the points on the graph of a function where the function isn’t differentiable.
3. * Practice: Differentiability at a point: graphical. (4 problems)
4. Differentiability at a point: algebraic (5:01) Is the function continuous/differentiable at ? (Optional)
5. Differentiability at a point: algebraic (6:21) Is the function continuous/differentiable at ? (Optional)
6. * Practice: Differentiability at a point: algebraic. (4 problems) (Optional)