1. The Natural Logarithmic Function: Differentiation (5.1)
December 6th, 2016

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 expression.

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

9. Ex. 2: Differentiate each function
Ex. 2: Differentiate each function. Properties of logarithms to expand the expression before differentiating, when beneficial. a. b. c. d.

10. Ex. 3: Find the relative extrema of .

11. *We can use logarithmic differentiation to differentiate nonlogarithmic functions.
Ex. 4: Use logarithmic differentiation to find the derivative of