1. 5.1 The Natural Logarithmic Function: Differentiation AB and BC 2015

2. Calculus Warm-Up 5.1 Find the derivative with respect to x:

3. Calculus Warm-Up 5.1 Find the indefinite integral:

6. Expanding Logarithmic Expressions

8. The Number e

9. Evaluating Natural Logarithmic Expressions

10. Derivative of ln x: The natural log function as follows: Now differentiate implicitly with respect to x: Rewrite in exponential form: Derivative of ln x

12. Example 1 – Differentiation of Logarithmic Functions

14. You Try:

15. Example 2 – Differentiation of Logarithmic Functions

16. Example 3 – You Try Find the derivative of the function:

17. If u > 0, If u < 0,

18. Example 4 – Derivative Involving Absolute Value You Try: Find the derivative of f(x) = ln | cosx |. Solution:

19. Example 5 – Derivative Involving Absolute Value You Try: Find the derivative of f(x) = ln | secx+tanx |.

20. Finding Relative Extrema Locate the Relative Extrema of:

21. Sometimes, it is convenient to use logarithms as aids in differentiating nonlogarithmic functions. This procedure is called logarithmic differentiation. Logarithmic Differentiation (Optional)

22. Example 6 – Logarithmic Differentiation Find the derivative of Solution: Note that y > 0 for all x ≠ 2. So, ln y is defined. Begin by taking the natural logarithm of each side of the equation. Then apply logarithmic properties and differentiate implicitly. Finally, solve for y'.

23. Example 6 – Solution cont’d

24. 5.1 AB Homework Page 330 45-59 odd, 63, 71, 77, 79, 83, 85

25. 5.1 BC Homework Day 1: Pg. 330 45-59 odd, 63, 71, 77, 79, 83, 85, 93-97 odd Day 2: MMM pgs.191-192