# Dx dy dx d 3 x 2 13 x 4 x 2 2 3 x 2 1 dx d 3 x 4 x 2

• 73

This preview shows page 30 - 33 out of 73 pages.

dx dy = dx d [(3 x 2 - 1)(3 x 4 - x 2 - 2)] = (3 x 2 - 1) dx d (3 x 4 - x 2 - 2) + (3 x 4 - x 2 - 2) dx d (3 x 2 - 1) = (3 x 2 - 1)(12 x 3 - 2 x ) + (3 x 4 - x 2 - 2) 6 x = 36 x 5 - 12 x 3 - 6 x 3 + 2 x + 18 x 5 - 6 x 3 - 12 x = 54 x 5 - 24 x 3 - 10 x 25. dx dy = + + - 1 2 4 2 x x x dx d = 2 2 2 ) 1 2 ( ) 1 2 ( ) 4 ( ) 4 ( ) 1 2 ( + + + - - + - + x x dx d x x x x dx d x = 2 2 ) 1 2 ( 2 ) 4 ( ) 1 2 )( 1 2 ( + + - - - + x x x x x = 2 2 2 ) 1 2 ( 8 2 2 1 4 + - + - - x x x x = 2 2 ) 1 2 ( 9 2 2 + - + x x x 26. dx dy = - + - 1 1 2 2 x x x dx d = 2 2 2 2 2 2 ) 1 ( ) 1 ( ) 1 ( ) 1 ( ) 1 ( - + - + - - - - + x x x x dx d x x dx d x x = 2 2 2 2 ) 1 ( ) 1 2 )( 1 ( 2 ) 1 ( - + + - - - + x x x x x x x = 2 2 2 3 2 3 ) 1 ( 1 2 2 2 2 2 - + + - + - - + x x x x x x x x = 2 2 2 ) 1 ( 1 - + + x x x 27. f ( x ) = 4 9 ) 2 3 )( 2 3 ( 2 + - x x x = 4 9 4 9 2 2 + - x x f ( x ) = + - 4 9 4 9 2 2 x x dx d = 2 2 2 2 2 2 ) 4 9 ( ) 4 9 ( ) 4 9 ( ) 4 9 ( ) 4 9 ( + + - - - + x x dx d x x dx d x = 2 2 2 2 ) 4 9 ( 18 ) 4 9 ( 18 ) 4 9 ( + - - + x x x x x = 2 2 3 3 ) 4 9 ( 72 162 72 162 + + - + x x x x x = 2 2 ) 4 9 ( 144 + x x 3 1 f = 2 2 4 3 1 9 3 1 144 + = 25 48 28. f ( x ) = ) 2 )( 1 2 ( ) 1 ( 2 + - - x x x x = 2 3 2 2 3 - + - x x x x f ( x ) = - + - 2 3 2 2 3 x x x x dx d = 2 2 2 3 3 2 ) 2 3 2 ( ) 2 3 2 ( ) ( ) ( ) 2 3 2 ( - + - + - - - - + x x x x dx d x x x x dx d x x = 2 2 3 2 2 ) 2 3 2 ( ) 3 4 )( ( ) 1 3 )( 2 3 2 ( - + + - - - - + x x x x x x x x = 2 2 3 2 4 2 2 3 4 ) 2 3 2 ( 3 3 4 4 2 3 2 6 9 6 - + + - + - + - - - + x x x x x x x x x x x = 2 2 2 3 4 ) 2 3 2 ( 2 4 6 2 - + + - + x x x x x f (0) = 2 2 2 3 4 ] 2 ) 0 ( 3 ) 0 ( 2 [ 2 ) 0 ( 4 ) 0 ( 6 ) 0 ( 2 - + + - + = 2 1
158 Solutions © Oxford University Press 2014 29. dx dy = dx d [ x (2 - x ) 3 ] = ) ( ) 2 ( ) 2 ( 3 3 x dx d x x dx d x - + - = 1 ) 2 ( ) 2 ( ) 2 ( 3 3 2 - + - - x x dx d x x = x 3(2 - x ) 2 ( - 1) + (2 - x ) 3 = (2 - x ) 2 ( - 3 x + 2 - x ) = (2 - x ) 2 ( - 4 x + 2) = - 2(2 - x ) 2 (2 x - 1) 30. dx dy = dx d [(3 x + 1) 4 ( x - 3)] = (3 x + 1) 4 dx d ( x - 3) + ( x - 3) dx d (3 x + 1) 4 = (3 x + 1) 4 1 + ( x - 3) 4(3 x + 1) 3 dx d (3 x + 1) = (3 x + 1) 4 + ( x - 3) 4(3 x + 1) 3 3 = (3 x + 1) 3 [3 x + 1 + 12( x - 3)] = (3 x + 1) 3 (15 x - 35) = 5(3 x + 1) 3 (3 x - 7) 31. dx dy = ) 4 5 ( 2 - x x dx d = ] ) 4 5 ( [ 2 1 2 - x x dx d = ) ( ) 4 5 ( ) 4 5 ( 2 1 2 2 1 2 x dx d x x dx d x - + - = 1 ) 4 5 ( ) 4 5 ( ) 4 5 ( 2 1 2 1 2 2 2 1 2 - + - - - x x dx d x x = 2 1 2 2 1 2 ) 4 5 ( 10 ) 4 5 ( 2 1 - + - - x x x x = ) 4 5 5 ( ) 4 5 ( 2 2 2 1 2 - + - - x x x = 4 5 4 10 2 2 - - x x = 4 5 ) 2 5 ( 2 2 2 - - x x 32. dx dy = + x x dx d 5 ) 7 2 ( = 2 5 5 ) ( ) 7 2 ( ) 7 2 ( x x dx d x x dx d x + - + = 2 5 4 1 ) 7 2 ( ) 7 2 ( ) 7 2 ( 5 x x x dx d x x + - + + = 2 5 4 ) 7 2 ( 2 ) 7 2 ( 5 x x x x + - + = 2 4 )] 7 2 ( 10 [ ) 7 2 ( x x x x + - + = 2 4 ) 7 8 ( ) 7 2 ( x x x - + 33. dx dy = + - 1 ) 3 ( 6 x x dx d = 2 6 6 ) 1 ( ) 1 ( ) 3 ( ) 3 ( ) 1 ( + + - - - + x x dx d x x dx d x = 2 6 5 ) 1 ( 1 ) 3 ( ) 3 ( ) 3 ( 6 ) 1 ( + - - - - + x x x dx d x x = 2 6 5 ) 1 ( ) 3 ( ) 1 ( ) 3 ( 6 ) 1 ( + - - - - + x x x x = 2 5 ) 1 ( )] 3 ( ) 1 ( 6 [ ) 3 ( + - - + - - x x x x = 2 5 ) 1 ( ) 3 6 6 ( ) 3 ( + + - - - - x x x x = 2 5 ) 1 ( ) 9 5 ( ) 3 ( + - - - x x x = 2 5 ) 1 ( ) 9 5 ( ) 3 ( + + - - x x x
Chapter 4: Differentiation 159 © Oxford University Press 2014 34. dx dy = - 1 2 2 x x dx d = - 2 1 2 ) 1 2 ( x x dx d = 2 2 1 2 2 1 2 2 1 2 ] ) 1 2 [( ) 1 2 ( ) ( ) 1 2 ( - - - - x x dx d x x dx d x = 1 2 ) 1 2 ( ) 1 2 ( 2 1 1 ) 1 2 ( 2 2 2 1 2 2 1 2 - - - - - - x x dx d x x x = 1 2 4 ) 1 2 ( 2 1 ) 1 2 ( 2 2 1 2 2 1 2 - - - - - x x x x x = 1 2 ) 2 1 2 ( ) 1 2 ( 2 2 2 2 1 2 - - - - - x x x x = 2 3 2 ) 1 2 ( 1 - - x 35. f ( x ) = ] 1 ) 1 ( [ 4 + - x dx d = 2 1 4 ] 1 ) 1 [( + - x dx d = ] 1 ) 1 [( ] 1 ) 1 [( 2 1 4 2 1 4 + - + - - x dx d x = ) 1 ( ) 1 ( 4 ] 1 ) 1 [( 2 1 3 2 1 4 - - + - - x dx d x x = 1 ) 1 ( 4 ] 1 ) 1 [( 2 1 3 2 1 4 - + - - x x = 1 ) 1 ( ) 1 ( 2 4 3 + - - x x f (0) = 1 ) 1 0 ( ) 1 0 ( 2 4 3 + - - = 2 2 - = 2 - 36. f ( x ) = dx d [( x + 1)(3 x - 1)] 5 = 5[( x + 1)(3 x - 1)] 4 dx d [( x + 1)(3 x - 1)] = 5[( x