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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 Â The Derivative as a Function Â

**Textbook Assignment**

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

**WeBWorK** **Assginment**

- Derivatives-Functions

**Exit problemsÂ ****of the sessionÂ **

- Let $f(x)=-x^2+2x}$. Â Use the definition of a derivative to find $f'(x)$.Â
- Let $g(x)=\dfrac{3}{x}$. Â Use the definition of a derivative to find $g'(x)$.Â

**Â Key Concepts**

- The derivative of a function $f(x)$ is the function whose value at $x$ is $f'(x)$:

$\displaystyle f'(x)=\lim_{h\to 0}\dfrac{f(x+h)-f(x)}{h}$

- If a function is differentiable at a point, then it is continuous at that point. Â 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 $n^{th}$ derivative.

#### Videos and Practice Problems of Selected Topics

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

- $\rhd$ Formal definition of the derivative as a limitÂ (15:42) The derivative of function $f$ at $x=x_0$ is the limit of the slope of the secant line from $x=x_0$ to $x=x_0+h$Â as $h$ approaches $0$.
- $\rhd$ The derivative of $x^2$ at any point using the formal definitionÂ (11:04)
- * Practice: Finding tangent lines using the formal definition of a limit. (6 questions with a guiding text)
- $\rhd$ Worked example: derivative as a limitÂ (5:45) Find $f'(e)$ when $f(x)=\ln(x)$.
- $\rhd$ Worked example: derivative from limit expressionÂ (5:16)Â Interpreting a limit expression as the derivative of $f(x)=x^3$ at the point $x=5$.
- * Practice: Â Derivative as a limit. (4 problems)
- * Derivative notation: a review of three notations. (two problems with a guiding text)

**Connecting differentiability and continuity**

- $\rhd$ Differentiability and ContinuityÂ (9:37) Defining differentiability and getting an intuition for the relationship between differentiability and continuity.
- $\rhd$ Differentiability at a point: graphical (5:38) FindÂ the points on the graph of a function where the function isn’t differentiable.
- * Practice: Differentiability at a point: graphical. (4 problems)
- $\rhd$ Differentiability at a point: algebraic (5:01) Is the function $f(x) = \begin{cases} x^2, & \text{ if } x<3\\ 6x-9, &\text{ if } x\geq 3\end{cases}$ continuous/differentiable at $x=3$?
*(Optional)* - $\rhd$ Differentiability at a point: algebraic (6:21) Is the function $g(x) = \begin{cases} x-1, & \text{ if } x<1\\ (x-1)^2, &\text{ if } x\geq 1\end{cases}$ continuous/differentiable at $x=1$?
*(Optional)* - * Practice: Differentiability at a point: algebraic. (4 problems)Â
*(Optional)*