differentiationoflogarithmicfunctions

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5.5 Differentiation of Logarithmic  Functions By Dr. Julia Arnold and Ms. Karen Overman using Tan’s 5th edition Applied Calculus for the managerial , life,  and social sciences text
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Now we will find derivatives of logarithmic functions and we will  Need rules for finding their derivatives. Rule 3:  Derivative of ln x ( 29 0 x Let’s see if we can discover why the rule is as above. ( 29 x y ln = First define the natural log function as follows: x e y = Now differentiate implicitly:  x 1 e 1 y 1 y e y y = = = Now rewrite in exponential form: x 1 x dx d = ln
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Example 1:   Find the derivative of f(x)= xlnx. Solution: This derivative will require the product rule.   1 lnx x 1 x (x) f xlnx f(x) + = = lnx 1 (x) f + = Product Rule: (1 st )(derivative of 2 nd ) + (2 nd )(derivative of 1 st )
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Example 2:   Find the derivative of g(x)= lnx/x Solution: This derivative will require the quotient rule. 2 x 1 lnx x 1 x (x) g x lnx g(x) - = = Quotient Rule: (bottom)(derivative of top) – (top)(derivative of bottom)                                      (bottom)² 2 x lnx 1 (x) g - =
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Why don’t you try one: Find the derivative of y = x²lnx .  The derivative will require you to use the product rule. Which of the following is the correct? y’ = 2 y’ = 2xlnx y’ = x + 2xlnx
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No, sorry that is not the correct answer.  Keep in mind -  Product Rule: (1 st )(derivative of 2 nd ) + (2 nd )(derivative of 1 st ) Try again.  Return to previous slide.
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F’(x) = (1 st )(derivative of 2 nd ) + (2 nd )(derivative of 1 st ) Good work! Using the product rule: y’ = x²          + (lnx)(2x) y’ = x + 2xlnx This can also be written y’ = x(1+2lnx) x 1
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Rule 4:  The Chain Rule for Log Functions [ ] ) ( ) ( ) ( ln x f x f x
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This document was uploaded on 11/04/2011 for the course MME 512 at Miami University.

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differentiationoflogarithmicfunctions -...

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