Thurs 3/6

Class Updates

• End of Week 8: Halfway through semester with 8 weeks to go.

• Spring Break next week: No homework due.

Goals for Today

• Finish section 3.6.

• Start section 3.9 (Chain Rule Application and Word Problems).

Upcoming Due Dates

• Homework 3.3 and both parts of 3.4 due tonight (Quizzes also).

  • 3.3: Shortcut formulas for the derivatives of trig functions.

  • 3.4: Chain rule.

• Previous sections (3.1 and 3.2) will be included in quizzes implicitly.

Review Session

Implicit Differentiation & Logarithms

• Importance of understanding logarithms for derivatives.

• Three key rules to remember:

  • Product Rule: Logarithm of a product can be split into a sum.

  • Quotient Rule: Logarithm of a quotient can be expressed as a difference.

  • Power Rule: Exponents can be moved in front of the logarithm.

• Avoid breaking up sums of logarithms (e.g., avoid simplifications like log(m + n)).

Example Problem Review

Logarithmic Differentiation Technique

• Illustration of Logarithmic Differentiation:

  • Function: H(x) = ln(x^x * (x + 1)²).

  • Break down using logarithmic properties:

    • H(x) = ln(x^x) + ln((x + 1)²).

  • Simplify each term and use derivatives:

    • d/dx[ln(f(x))] = 1/f(x) * f'(x).

Step-by-Step Procedure for Example 3

1. Rewrite H(x) to express as sums.

2. Derivative involves using product and chain rules.

3. Evaluate each component by applying the implicit differentiation techniques learned.

Additional Notes on Related Rates (Upcoming Topic)

• Concept of Related Rates:

  • Derivatives of related quantities in a situation where they are related through a function.

  • E.g., how the volume of a balloon changes with respect to its radius.

• Procedure:

  1. Relate the variables with an appropriate equation (e.g., volume of a sphere).

  2. Identify given information and desired outputs.

  3. Differentiate to find derivatives and evaluate at specified conditions (e.g., at specific radius values).

Preparing for Spring Break

• Suggestion: Work on derivative packets to reinforce skills.

• Use this downtime for any missed assignments on WebAssign.