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Derivatives of Logarithmic Functions
Section 3.6
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Example 1, differentiate
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Example 2, differentiate
Find ln(sin x). Solution: Using the chain rule we have
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Example 3, differentiate
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Example 4, differentiate
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Example 5, differentiate
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Example 5, differentiate
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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.
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Example 6, differentiate
Solution: Take log of both sides of the equation Use the properties of logarithms to simplify:
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Example 6, differentiate
Use implicit differentiation:
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Example 6, differentiate
Substitute y back in:
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3.6 Derivatives of Logarithmic Functions
Summarize Notes Read section 3.6 Homework Pg.223 #2-32 (odd)
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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
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Because f (1) = 1, we have Then, by the continuity of the exponential function, we have
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