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Derivatives of Logarithmic Functions

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Presentation on theme: "Derivatives of Logarithmic Functions"— Presentation transcript:

1 Derivatives of Logarithmic Functions
Section 3.6

2

3 Example 1, differentiate

4 Example 2, differentiate
Find ln(sin x). Solution: Using the chain rule we have

5 Example 3, differentiate

6 Example 4, differentiate

7 Example 5, differentiate

8 Example 5, differentiate

9 The calculation of derivatives of complicated functions involving products, quotients, or powers can often be simplified by taking logarithms. The method used in the next example is called logarithmic differentiation.

10 Example 6, differentiate
Solution: Take log of both sides of the equation Use the properties of logarithms to simplify:

11 Example 6, differentiate
Use implicit differentiation:

12 Example 6, differentiate
Substitute y back in:

13

14 3.6 Derivatives of Logarithmic Functions
Summarize Notes Read section 3.6 Homework Pg.223 #2-32 (odd)

15 The Number e as a Limit If f (x) = ln x, then f (x) = 1/x. Thus f (1) = 1. We now use this fact to express the number e as a limit From the definition of a derivative as a limit, we have

16 Because f (1) = 1, we have Then, by the continuity of the exponential function, we have


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