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Answer Guide 4 Math 408C: Unique Numbers 56975, 56980, and 56985 Tuesday, September 22, 2009 Homework problems Section 3.3 70. (a) If y ( x ) = x 2 f ( x ) , then y 0 ( x ) = x 2 f 0 ( x ) + 2 xf ( x ) : (b) If y ( x ) = f ( x ) =x 2 , then y 0 ( x ) = f 0 ( x ) x 2 2 xf ( x ) ( x 2 ) 2 = xf 0 ( x ) 2 f ( x ) x 3 = f 0 ( x ) x 2 2 f ( x ) x 3 : (c) If y ( x ) = x 2 =f ( x ) , then y 0 ( x ) = 2 xf ( x ) x 2 f 0 ( x ) [ f ( x )] 2 (d) If y ( x ) = [1 + xf ( x )] = p x , then y 0 ( x ) = d dx [1 + xf ( x )] p x [1 + xf ( x )] d dx ( p x ) ( p x ) 2 = xf 0 ( x ) + f ( x ) p x [1 + xf ( x )] x 1 2 x 1 = 2 = x 2 f 0 ( x ) + xf ( x ) 1 2 1 2 xf ( x ) x 3 = 2 = 2 x 2 f 0 ( x ) + xf ( x ) 1 2 x 3 = 2 : 1

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Section 3.4 8. If y ( u ) = u ( a cos u + b cot u ) , then y 0 ( u ) = a cos u + b cot u au sin u bu csc 2 u: 10. If y ( x ) = 1+sin x x +cos x , then y 0 ( x ) = ( x + cos x ) cos x (1 + sin x )(1 sin x ) ( x + cos x ) 2 = x cos x + cos 2 x (1 sin 2 x ) ( x + cos x ) 2 = x cos x ( x + cos x ) 2 : 14. If y ( ) = ( + cot ) csc , then y 0 ( ) = (1 csc 2 ) csc + ( + cot )( csc cot ) = csc (1 csc 2 cot cot 2 ) = csc
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