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Stewart6e3_6

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

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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 we’ve done with derivatives, we still don’t know how to differentiate logarithms. It’s 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: there’s 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 )

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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 there’s a function more general than ln( x ) (in that it’s defined on a larger domain and it agrees with ln( x ) on their common domain) with the same derivative.

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

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