Chapter_11.2 - Chapter 11.2 Page 595. ' (i) ex = ex ( )...

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Unformatted text preview: Chapter 11.2 Page 595. ' (i) ex = ex ( ) Example: (A) Find the derivatives for f ( x) = 4 x + 3e x ( ) 4 f ' ( x) = [ 4 x + 3e x ]' = 4 4 x + 3e x = 4 4 x + 3e (B) f ( x) = ex x3 ( ( ) 4 x 3 ) ( ) ( 4 x + 3e ) 3 x ' (4 + 3e x ) = (16 + 12e x ) 4 x + 3e x ( Chain _ Rule ) 3 f ' ( x) = (e x )' x 3 - e x ( x 3 )' Quotient _ Rule (x3 )2 = (e x ) x 3 - e x (3 x 2 ) e x x 2 ( x - 3) e x ( x - 3) = = x6 x6 x4 u ( x) ' (ii) (e ) = e u ( x ) u ' ( x) Find the derivatives for 2 Example: (A) f ( x) = e x 2 f ' ( x) = (e x )' = e x ( x 2 )' = 2 xe x (B) f ( x) = 10 + 5e -2 x 2 2 Rewrite f ( x) = 10 + 5e -2 x = 10 + 5e - 2 x f ' ( x) = [ 10 + 5e -2 x = ( ) 1 2 first! 10 + 5e - 2 x ( ) 1 2 ' = 1 10 + 5e -2 x 2 - 1 2 1 2 ( ) 1 -1 2 ( ) ' 1 10 + 5e -2 x 2 1 = 10 + 5e -2 x 2 ( ( ) ) (10)'+ (5e -2 x ) ' = ( ( ) 1 10 + 5e - 2 x 2 ( ) ) - 1 2 5 -2e - 2 x = - 5e -2 x ( ) - - 10e -2 x = -5e -2 x 10 + 5e -2 x ) ( - 1 2 (10 - 5e ) -2 x 1 Page 597. (i) ( ln x ) ' = 1 ; x (ii) ( ln u ( x) ) ' = 1 u ' ( x) ; u ( x) Page 736. Example: (A) Find the derivatives for 4 f ( x) = ( ln x ) 4 1 4( ln x ) f ' ( x) = [( ln x ) ]' = 4 ( ln x ) ( ln x ) = 4( ln x ) = x x 3 ' 3 3 (B) y = x 2 ln x y ' = ( x 2 ln x)' = ( x 2 )' ln x + x 2 (ln x )' Product Rule 1 = (2 x ) ln x + x 2 = 2 x ln x + x x Example: (A) Find the derivatives for g ( x) = ln(1 + x 2 ) 1 2x (1 + x 2 )' = 2 (1 + x ) (1 + x 2 ) g ' ( x ) = [ln(1 + x 2 )]' = (B) y = ln x 4 y ' = (ln x 4 )' = Page 598. Remark: 1 1 4 ( x 4 )' = 4 (4 x 3 ) = 4 x x x Change of basis ln b ln a y = log a b if and only if y = 2 Example: (A) Find the derivatives for f ( x) = log 2 x ln x ln 2 Rewrite f ( x) = log 2 x = ' 1 1 1 1 ln x f ' ( x) = (ln x )' = = = ln 2 x x ln 2 ln 2 ln 2 (B) f ( x) = log(1 + x 3 ) , here the base is 10 ln(1 + x 3 ) ln 10 Rewrite f ( x) = log(1 + x 3 ) = ' ln(1 + x 3 ) 1 1 1 3 ' 3 f ' ( x) = ln 10 = ln 10 ln(1 + x ) = ln 10 (1 + x 3 ) (1 + x )' Chain Rule 1 1 3x 2 = (3 x 2 ) = ln 10 (1 + x 3 ) (1 + x 3 ) ln 10 ( ) Additional Examples: Find the derivative for Example 1: f ( x) = x( ln x ) 2 1 2 2 f ' ( x) = ( x)' ( ln x ) + x[(ln x) 2 ]' = ( ln x ) + x [2 ln x ] = (ln x) 2 + 2 ln x x f ( x) = (5 - ln x) 4 Example 2: f ' ( x) = [(5 - ln x) 4 ]' = 4 (5 - ln x) 3 (5 - ln x) ' Chain Rule 1 4(5 - ln x ) 3 = 4 (5 - ln x) 3 (- ) = - x x Example 3: f ( x) = x ln x - x 1 - 1 = ln x + 1 - 1 = ln x x f ' ( x) = ( x ln x )'-( x)' = ( x)' ln x + x(ln x )'-1 = ln x + x 3 ...
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