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

 

 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.  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.
 

Videos and Practice Problems of Selected Topics

  • The definition of the derivative as a limit
  1. $\rhd$ Formal definition of the derivative as a limit (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 $x=x_0+h$ as $h$ approaches $0$.
  2. $\rhd$ The derivative of $x^2$ 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. $\rhd$ Worked example: derivative as a limit (5:45) Find $f'(e)$ when $f(x)=\ln(x)$.
  5. $\rhd$ Worked example: derivative from limit expression (5:16) Interpreting a limit expression as the derivative of $f(x)=x^3$ at the point $x=5$.
  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. $\rhd$ Differentiability and Continuity (9:37) Defining differentiability and getting an intuition for the relationship between differentiability and continuity.
  2. $\rhd$ 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. $\rhd$ 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) (Optional)