Aug 5, 2024

- Purpose of lecture:
- To explain the motivation behind the chain rule.
- To clarify the significance of the exercise.

- Initial thoughts on differentiation:
- Difficulty of term-by-term differentiation.
- Need for a more efficient method.

- Example presented:
- Simplistic appearance of a graph, yet differentiation can be impractical.

- Importance of rewriting functions:
- Example: Rewriting the square root function for clarity.

- Steps in differentiation using the chain rule:
- Identify the inside function (e.g.,
`4 - x`

). - Differentiate the inside function:
- Derivative of
`4 - x`

is`-1`

.

- Derivative of
- Identify the outside function:
- E.g., a power function (e.g.,
`1/2`

for square root).

- E.g., a power function (e.g.,
- Apply the power rule:
- Bring the power out as a coefficient.
- Reduce the power by one.

- Identify the inside function (e.g.,

- Tidying up the results:
- Convert to a fraction to avoid index form.
- Example: Resulting in
`-1/(2√(4 - x))`

.

- Understanding the derivative's significance:
- Derivative represents the gradient function.

- Visualizing the function:
- Derivative resembles a sideways parabola.
- The negative sign reflects a horizontal flip.

- Characteristics of the function:
- X-intercept at
`4`

and Y-intercept at`2`

.

- X-intercept at

- Insights from the derivative:
- Square root in the denominator indicates positivity.
- Overall derivative is negative, indicating a decreasing function.

- Derivative defined for:
`x < 4`

(not inclusive due to the denominator).

- Asymptotic behavior:
- Presence of an asymptote at
`x = 4`

due to the denominator.

- Presence of an asymptote at