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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 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

1. Let $f(x)=-x^2+2x}$. Â Use the definition of a derivative to find $f'(x)$.Â
2. 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 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 $n^{th}$ derivative.
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#### Videos and Practice Problems of Selected Topics

• The definition of the derivative as a limit
1. $\rhd$ 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 $0$.
2. $\rhd$ The derivative of $x^2$ at any point using the formal definitionÂ (11:04)
3. * Practice: Finding tangent lines using the formal definition of a limit. (6 questions with a guiding text)
4. $\rhd$ Worked example: derivative as a limitÂ (5:45) Find $f'(e)$ when $f(x)=\ln(x)$.
5. $\rhd$ Worked example: derivative from limit expressionÂ (5:16)Â Interpreting a limit expression as the derivative of $f(x)=x^3$ at the point $x=5$.
6. * Practice: Â 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. $\rhd$ Differentiability and ContinuityÂ (9:37) Defining differentiability and getting an intuition for the relationship between differentiability and continuity.
2. $\rhd$ Differentiability at a point: graphical (5:38) FindÂ the points on the graph of a function where the function isn’t differentiable.
3. * Practice: Differentiability at a point: graphical. (4 problems)
4. $\rhd$ 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)
5. $\rhd$ 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)
6. * Practice: Differentiability at a point: algebraic. (4 problems)Â (Optional)