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practice2sol

# practice2sol - MAT 21A 1(42 pts Fu Liu Practice Exam 2...

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MAT 21A Fu Liu Practice Exam 2 1 (42 pts.) Differentiate the following functions. Show your work if you wish to receive partial credit. You do not need to simplify your answers. (The step where I marked with a is the place where you can finish as an answer.) (a) f ( x ) = 3 x +7 x 4 +1 . Solution : f ( x ) = x 4 + 1 d dx (3 x + 7) - (3 x + 7) d dx ( x 4 + 1) x 4 + 1 = 3 x 4 + 1 - (3 x + 7) 1 2 x 4 +1 d dx ( x 4 + 1) x 4 + 1 = 3 x 4 + 1 - (3 x + 7) 2 x 3 x 4 +1 x 4 + 1 = - 3 x 4 - 14 x 3 + 3 ( x 4 + 1) 3 / 2 (b) f ( x ) = sec 2 ( x + sin x ) . Solution : f ( x ) = 2 sec( x + sin x ) d dx (sec( x + sin x )) = 2 sec( x + sin x ) sec( x + sin x ) tan( x + sin x ) d dx ( x + sin x ) = 2 sec 2 ( x + sin x ) tan( x + sin x )(1 + cos x ) (c) f ( x ) = e 2 x - 1 x + 1 . Solution : f ( x ) = e 2 x - 1 d dx ( x + 1) + x + 1 d dx ( e 2 x - 1 ) = e 2 x - 1 1 2 x + 1 d dx ( x + 1) + x + 1 e 2 x - 1 d dx (2 x - 1) = e 2 x - 1 1 2 x + 1 + 2 x + 1 e 2 x - 1 = e 2 x - 1 4 x + 5 2 x + 1

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(d) f ( x ) = 1 (ln x ) 10 . Solution : f ( x ) = (ln x ) - 10 . f ( x ) = - 10 (ln x ) - 11 d dx (ln x ) = - 10 (ln x ) - 11 1 x = - 10 x (ln x ) 11 (e) f ( x ) = x 2 +1 tan - 1 ( x 2 +1) . Solution : f ( x ) = (tan - 1 ( x 2 + 1)) d dx ( x 2 + 1) - ( x 2 + 1) d dx (tan - 1 ( x 2 + 1)) (tan - 1 ( x 2 + 1)) 2 = 2 x (tan - 1 ( x 2 + 1)) - ( x 2 + 1) 1 ( x 2 +1) 2 +1 d dx ( x 2 + 1) (tan - 1 ( x 2 + 1)) 2 = 2 x (tan - 1 ( x 2 + 1)) - 2 x ( x 2
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practice2sol - MAT 21A 1(42 pts Fu Liu Practice Exam 2...

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