1. The Natural Logarithmic Function: Differentiation (5.1) February 21st, 2013

2. I. The Natural Logarithmic Function Def. of the Natural Logarithmic Function: The natural logarithmic function is defined by. The domain is the set of all positive real numbers.

3. Thm. 5.1: Properties of the Natural Logarithmic Function: 1. Domain:, Range: 2. The function is continuous, increasing, and one- to-one 3. The graph is concave downward

4. Thm. 5.2: Logarithmic Properties: 1. 2. 3. 4.

5. Ex. 1: Use properties of logarithms to expand the following logarithmic expressions. a. b.

6. II. The Number e *Recall that the base of the natural logarithm is the number, so. Def. of e: The letter e denotes the positive real number such that.

7. III. The Derivative of the Natural Logarithm Thm. 5.3: Derivative of the Natural Logarithmic Function: Let u be a differentiable function of x. 1. (since ) 2.

8. Ex. 2: Differentiate each function. a. b. c. d. e. f.

9. *We can use logarithmic differentiation to differentiate nonlogarithmic functions. Ex. 3: Use logarithmic differentiation to find the derivative of.

10. You Try: Use logarithmic differentiation to find the derivative of.

11. Thm. 5.4: Derivative Involving Absolute Value: If u is a differentiable function of x such that, then.

12. Ex. 4: Find the derivative of.

13. Ex. 5: Find the relative extrema of.