- The definition of the derivative as a limit
- $\rhd$ 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 $0$.
- $\rhd$ The derivative of $x^2$ 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)
- $\rhd$ Worked example: derivative as a limit (5:45) Find $f'(e)$ when $f(x)=\ln(x)$.
- $\rhd$ Worked example: derivative from limit expression (5:16) Interpreting a limit expression as the derivative of $f(x)=x^3$ at the point $x=5$.
- * Practice: 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 Continuity (9:37) Defining differentiability and getting an intuition for the relationship between differentiability and continuity.
- $\rhd$ 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)
- $\rhd$ 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 $x=3$?
- $\rhd$ 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 $x=1$?
- * Practice: Differentiability at a point: algebraic. (4 problems)
