Derivatives of Logarithmic Functions Section 3.6

Example 1, differentiate

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

Example 3, differentiate

Example 4, differentiate

Example 5, differentiate

Example 5, differentiate

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.

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

Example 6, differentiate Use implicit differentiation:

Example 6, differentiate Substitute y back in:

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

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

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