Stewart6e3_6 - MATH 191, Sections 1 and 3 Calculus I Fall...

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Unformatted text preview: MATH 191, Sections 1 and 3 Calculus I Fall 2007 Section 3.6: Derivatives of Logarithmic Functions The astute observer may have noticed that with all weve done with derivatives, we still dont know how to differentiate logarithms. Its about time we learned! Let y = log a ( x ), so that a y = x . Go ahead and use the space below to find the derivative d dx ( log a ( x ) ) by implicit differentiation: Thus, in particular, d dx ln( x ) = 1 ln( e ) x = 1 x . (Notice that this fills neatly a gap that appeared in the Power Rule: theres no way we could have obtained 1 x = x- 1 as a derivative, since d dx c = 0 for constants c . Combining our newfound knowledge with the Chain Rule, we get tons o fun! Examples. Differentiate the following functions: 1. f ( x ) = ln( x 3 + 2 x ) 2. g ( x ) = ln(cos( x )) 3. H ( t ) = log 2 ( t 2 ) 4. f ( x ) = (ln( x )) 5 5. f ( x ) = ln( | x | ) The last example is an important one for later purposes: we see that theres a function more general...
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This note was uploaded on 04/08/2008 for the course MATH 191 taught by Professor Bahls during the Fall '07 term at UNC Asheville.

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Stewart6e3_6 - MATH 191, Sections 1 and 3 Calculus I Fall...

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