Deck 4: More About Derivatives

Question 1

Let $y = 5 \left( x ^ { 2 } - x + 1 \right) ^ { 100 }$ Then $y ^ { \prime }$ is

A) $500 ( 2 x + 1 ) ^ { 99 }$
B) $500 ( 2 x - 1 ) ^ { 99 }$
C) $1000 \left( x ^ { 2 } - x + 11 \right) ^ { 99 }$
D) $500 \left( x ^ { 2 } - x + 1 \right) ^ { 99 } ( 2 x + 1 )$
E) $500 \left( x ^ { 2 } - x + 1 \right) ^ { 99 } ( 2 x - 1 )$

Answer 1

$500 \left( x ^ { 2 } - x + 1 \right) ^ { 99 } ( 2 x - 1 )$

Question 2

Let $y = \cos \left( x ^ { 2 } - x + 1 \right)$ . Then $y ^ { \prime }$ is

A) $- \sin ( 2 x - 1 )$
B) $\sin ( 2 x - 1 )$
C) $( 2 x - 1 ) \sin \left( x ^ { 2 } - x + 1 \right)$
D) $- ( 2 x - 1 ) \sin \left( x ^ { 2 } - x + 1 \right)$
E) $- 2 x \sin \left( x ^ { 2 } - x + 1 \right)$

Answer 2

$- ( 2 x - 1 ) \sin \left( x ^ { 2 } - x + 1 \right)$

Question 3

Let $y = \tan ^ { 3 } x$ Then $y ^ { \prime }$ is

A) $3 \tan ^ { 2 } x$
B) $3 \tan x \sec ^ { 2 } x$
C) $3 \tan ^ { 2 } x \sec ^ { 2 } x$
D) $3 \sec ^ { 2 } x$
E) $3 \tan ^ { 2 } x \sec ^ { 4 } x$

Answer 3

$3 \tan ^ { 2 } x \sec ^ { 2 } x$

Question 4

Let $y = \csc ( 5 x ) .$ Then $y ^ { \prime }$ is

A) $5 \csc ( 5 x ) \cot ( 5 x )$
B) $- 5 \csc ( 5 x ) \cot ( 5 x )$
C) $5 \csc x \cot x$
D) $- 5 \csc x \cot x$
E) $- 5 \csc ^ { 2 } x$

Question 5

Let $y = \frac { u } { 1 - u }$ and $u = 2 x$ By the Chain Rule, $\frac { d y } { d x }$ is

A) $\frac { 1 } { ( 1 - x ) ^ { 2 } }$
B) $\frac { 2 } { ( 1 - 2 x ) ^ { 2 } }$
C) $\frac { x } { ( 1 - 2 x ) ^ { 2 } }$
D) $\frac { 2 x } { ( 1 - 2 x ) ^ { 2 } }$
E) $- \frac { x } { ( 1 - 2 x ) ^ { 2 } }$

Question 6

Let $y = \frac { u - 2 } { u + 3 }$ and $u = 2 x + 1$ By the Chain Rule, $\frac { d y } { d x }$ is

A) $\frac { 4 x + 7 } { ( 2 x + 4 ) ^ { 2 } }$
B) $\frac { 4 x - 7 } { ( 2 x + 4 ) ^ { 2 } }$
C) $\frac { 10 } { ( 2 x + 4 ) ^ { 2 } }$
D) $- \frac { 5 } { ( 2 x + 4 ) ^ { 2 } }$
E) $\frac { 5 } { ( 2 x + 4 ) ^ { 2 } }$

Question 7

Let $y = u + \frac { 1 } { u }$ and $u = x ^ { 2 } + x - 1$ By the Chain Rule, $\frac { d y } { d x }$ is

A) $1 + \frac { 1 } { x ^ { 2 } + x - 1 }$
B) $1 - \frac { 1 } { x ^ { 2 } + x - 1 }$
C) $1 - \frac { 1 } { \left( x ^ { 2 } + x - 1 \right) ^ { 2 } }$
D) $( 2 x + 1 ) \left( 1 - \frac { 1 } { \left( x ^ { 2 } + x - 1 \right) ^ { 2 } } \right)$
E) $( 2 x + 1 ) \left( 1 + \frac { 1 } { \left( x ^ { 2 } + x - 1 \right) ^ { 2 } } \right)$

Question 8

Let $y = \sin u + \cos u$ and $u = 4 x ^ { 2 } + 5$ By the Chain Rule, $\frac { d y } { d x }$ is

A) $\cos ( 8 x ) - \sin ( 8 x )$
B) $\cos ( 8 x ) + \sin ( 8 x )$
C) $\cos \left( 4 x ^ { 2 } + 5 \right) - \sin \left( 4 x ^ { 2 } + 5 \right)$
D) $\cos \left( 4 x ^ { 2 } + 5 \right) + \sin \left( 4 x ^ { 2 } + 5 \right)$
E) $8 x \left[ \cos \left( 4 x ^ { 2 } + 5 \right) - \sin \left( 4 x ^ { 2 } + 5 \right) \right]$

Question 9

Let $y = \frac { u } { e ^ { u } }$ and $u = \frac { 3 } { x } .$ By the Chain Rule, $\frac { d y } { d x }$ is

A) $\frac { x - 9 } { x ^ { 2 } e ^ { \frac { 3 } { x } } }$
B) $\frac { x - 3 } { x ^ { 3 } e ^ { \frac { 3 } { x } } }$
C) $\frac { 3 ( 3 - x ) } { x ^ { 3 } e ^ { \frac { 3 } { x } } }$
D) $\frac { 9 - x } { x ^ { 2 } e ^ { \frac { 3 } { x } } }$
E) $\frac { 9 - x } { x ^ { 3 } e ^ { \frac { 3 } { x } } }$

Question 10

Let $y = 3 u e ^ { u }$ and $u = 2 x + 3$ By the Chain Rule, $\frac { d y } { d x }$ is

A) $3 e ^ { 2 x + 3 }$
B) $6 e ^ { 2 x + 3 } ( 2 x + 3 )$
C) $12 e ^ { ( 2 x + 3 ) } ( x + 2 )$
D) $12 e ^ { ( 2 x + 3 ) } ( 2 x + 3 )$
E) $6 e ^ { 2 x + 3 }$

Question 11

Let $f ( x ) = \sin ( \cos x )$ Then $f ^ { \prime } ( x )$ is

A) $\cos ( \sin x )$
B) $\cos ( \cos x )$
C) $\sin x [ \cos ( \cos x ) ]$
D) $- \sin x [ \cos ( \cos x ) ]$
E) $- \sin x [ \sin ( \cos x ) ]$

Question 12

Let $f ( x ) = \sin \left( e ^ { x } \right)$ Then $f ^ { \prime } ( x )$ is

A) $\cos \left( e ^ { x } \right)$
B) $e ^ { x } \cos \left( e ^ { x } \right)$
C) $- e ^ { x } \cos \left( e ^ { x } \right)$
D) $e ^ { x } \sin \left( e ^ { x } \right)$
E) $- e ^ { x } \sin \left( e ^ { x } \right)$

Question 13

Let $f ( x ) = \left( e ^ { x } + x ^ { e } \right) ^ { 6 }$ . Then $f ^ { \prime } ( x )$ is

A) $6 \left( e ^ { x } + x ^ { 6 } \right) ^ { 5 }$
B) $6 \left( x e ^ { x - 1 } + x ^ { \prime } \right) ^ { 5 }$
C) $6 \left( e ^ { x } + e x ^ { e - 1 } \right) ^ { 5 }$
D) $6 \left( e ^ { x } + e x ^ { e - 1 } \right) ^ { 6 }$
E) $6 \left( e ^ { x } + x ^ { e } \right) ^ { 5 } \left( e ^ { x } + e x ^ { e - 1 } \right)$

Question 14

Let $f ( x ) = \tan ^ { 3 } ( 5 x )$ Then $f ^ { \prime } ( x )$ is

A) $15 \tan ^ { 2 } ( 5 x )$
B) $15 \tan ^ { 2 } ( 5 x ) \sec ( 5 x )$
C) $15 \tan ^ { 2 } ( 5 x ) \sec ^ { 2 } ( 5 x )$
D) $5 \tan ^ { 3 } ( 5 x ) \sec ^ { 2 } ( 5 x )$
E) $5 \tan ^ { 2 } ( 5 x ) \sec ( 5 x )$

Question 15

Let $f ( x ) = \tan \left( 5 ^ { x } \right)$ Then $f ^ { \prime } ( x )$ is

A) $\tan \left( 5 ^ { x } \ln 5 \right)$
B) $\sec ^ { 2 } \left( 5 ^ { x } \right)$
C) $\sec ^ { 2 } \left( 5 ^ { x } \ln 5 \right)$
D) $\sec ^ { 2 } \left( 5 ^ { x } \right) \ln 5$
E) $5 ^ { x } \sec ^ { 2 } \left( 5 ^ { x } \right) \ln 5$

Question 16

Let $f ( x ) = \mathrm { e } ^ { \cos ( 3 x ) }$ Then $f ^ { \prime } ( x )$ is

A) $3 e ^ { \sin ( 3 x ) }$
B) $e ^ { 3 x } \sin ( 3 x )$
C) $3 e ^ { 3 x } \sin ( 3 x )$
D) $3 \sin ( 3 x ) \mathrm { e } ^ { \cos ( 3 x ) }$
E) $- 3 \sin ( 3 x ) \mathrm { e } ^ { \cos ( 3 x ) }$

Question 17

Let $f ( x ) = 5 ^ { \cot ( 3 x ) }$ Then $f ^ { \prime } ( x )$ is

A) $15 ^ { \csc ^ { 2 } ( 3 x ) }$
B) $- 15 ^ { \csc ^ { 2 } ( 3 x ) }$
C) $15 ^ { \cot ( 3 x ) } \left( \csc ^ { 2 } ( 3 x ) \right)$
D) $5 ^ { \cos ( 3 x ) } \left( 3 \csc ^ { 2 } ( 3 x ) \right)$
E) $- 5 ^ { \cos ( 3 x ) } \left( 3 \csc ^ { 2 } ( 3 x ) \ln 5 \right)$

Question 18

Let $f ( x ) = \cot \left( e ^ { x } \right)$ Then $f ^ { \prime } ( x )$ is

A) $- \csc ^ { 2 } \left( e ^ { x } \right)$
B) $- \cot ^ { 2 } \left( e ^ { x } \right)$
C) $- e ^ { x } \cot ^ { 2 } \left( e ^ { x } \right)$
D) $- e ^ { x } \csc ^ { 2 } \left( e ^ { x } \right)$
E) $e ^ { x } \csc ^ { 2 } \left( e ^ { x } \right)$

Question 19

Let $h = f \circ g .$ Then $g ( 2 ) = 3$ , $g ^ { \prime } ( 2 ) = - 1$ and $f ^ { \prime } ( 3 ) = 6$ Then $h ^ { \prime } ( 2 )$ is

A)-6
B)6
C)-3
D)18
E)16

Question 20

Let $h = g \circ f$ Then and $f ( 2 ) = 4$ , $f ^ { \prime } ( 2 ) = \frac { 1 } { 2 }$ and $g ^ { \prime } ( 4 ) = 10$ Then $h ^ { \prime } ( 2 )$ is

A)40
B)20
C)10
D)5
E)2

Question 21

Using implicit differentiation on $e ^ { y - x } = x , y ^ { \prime }$ is

A) $1 + e ^ { x - y }$
B) $1 + e ^ { y - x }$
C) $1 - e ^ { x - y }$
D) $e ^ { x - y } - 1$
E) $e ^ { y - x } - 1$

Question 22

Using implicit differentiation on $2 x ^ { 2 } - x y + y ^ { 2 } = 4 , y$ is

A) $\frac { y - 4 x } { x - 2 y }$
B) $\frac { y - 4 x } { x + 2 y }$
C) $\frac { 4 x - y } { x - 2 y }$
D) $\frac { 4 x - y } { x + 2 y }$
E) $\frac { 4 x + y } { x - 2 y }$

Question 23

Using implicit differentiation on $\sin y - \cos x + x y = 2 , y ^ { \prime }$ is

A) $- \frac { \sin x + y } { \cos y + x }$
B) $\frac { \sin x - y } { \cos y + x }$
C) $\frac { \sin x + y } { \cos y + x }$
D) $\frac { \sin x - y } { \cos y - x }$
E) $\frac { - \sin x + y } { \cos y + x }$

Question 24

Using implicit differentiation on $y = \cos ( x + y ) , y ^ { \prime }$ is

A) $\frac { \sin ( x + y ) } { 1 + \sin ( x + y ) }$
B) $\frac { \sin ( x + y ) } { 1 - \sin ( x + y ) }$
C) $\frac { \sin ( x + y ) } { \sin ( x + y ) - 1 }$
D)-1
E) $- \frac { \sin ( x + y ) } { 1 + \sin ( x + y ) }$

Question 25

Using implicit differentiation on $y = ( x + 2 y ) ^ { 2 } , y ^ { \prime }$ is

A) $\frac { x + 2 y } { 1 - 2 ( x + 2 y ) }$
B) $\frac { 2 ( x + 2 y ) } { 4 ( x + 2 y ) - 1 }$
C) $\frac { 2 ( x + 2 y ) } { 1 - 4 ( x + 2 y ) }$
D) $\frac { 2 ( x + 2 y ) } { 1 + 4 ( x + 2 y ) }$
E) $\frac { 2 ( x - 2 y ) } { 1 - 4 ( x + 2 y ) }$

Question 26

Using implicit differentiation on $x y - 5 ^ { x + y } = 8 , y ^ { \prime }$ is

A) $\frac { x - 5 ^ { x + y } \ln 5 } { 5 ^ { x + y } \ln 5 - y }$
B) $\frac { x + 5 ^ { x + y } \ln 5 } { 5 ^ { x + y } \ln 5 - y }$
C) $\frac { x + 5 ^ { x + y } \ln 5 } { 5 ^ { x + y } \ln 5 + y }$
D) $\frac { 5 ^ { x + y } \ln 5 - y } { x - 5 ^ { x + y } \ln 5 }$
E) $\frac { 5 ^ { x + y } - y } { x - 5 ^ { x + y } \ln 5 }$

Question 27

Using implicit differentiation on $y = x \sin y , y ^ { \prime }$ is

A) $\frac { \cos y } { 1 - x \sin y }$
B) $\frac { \cos y } { 1 + x \sin y }$
C) $\frac { \sin y } { 1 - x \cos y }$
D) $\frac { \sin y } { 1 + x \cos y }$
E) $\frac { \sin y } { x \cos y - 1 }$

Question 28

Using implicit differentiation on $x ^ { 2 } + y ^ { 2 } = 2 y , y ^ { \prime }$ is

A) $\frac { x } { 1 - y }$
B) $\frac { x } { y - 1 }$
C) $\frac { x } { 1 + y }$
D) $\frac { y } { 1 - x }$
E) $\frac { y } { x - 1 }$

Question 29

Using implicit differentiation on $x ^ { 3 } - 3 x y + y ^ { 3 } = 6 , y\prime$ is

A) $\frac { y ^ { 2 } - x } { y - x ^ { 2 } }$
B) $\frac { x - y ^ { 2 } } { y - x ^ { 2 } }$
C) $\frac { x ^ { 2 } - y } { y ^ { 2 } - x }$
D) $\frac { y - x ^ { 2 } } { y ^ { 2 } - x }$
E) $\frac { y + x ^ { 2 } } { y ^ { 2 } - x }$

Question 30

Using implicit differentiation on $e ^ { y } + \sin ( x - y ) = x ^ { 2 } , y ^ { \prime }$ is

A) $\frac { \cos ( x - y ) - 2 x } { e ^ { y } - \cos ( x - y ) }$
B) $\frac { 1 - 2 x } { e ^ { y } - 1 }$
C) $\frac { 2 - 1 x } { e ^ { y } - 1 }$
D) $\frac { 2 x - \cos ( x - y ) } { e ^ { y } - \cos ( x - y ) }$
E) $\frac { 2 x + \cos ( x - y ) } { e ^ { y } - \cos ( x - y ) }$

Question 31

Let $f ( x ) = x ^ { 2 } \sin ^ { - 1 } x$ Then $f ^ { \prime } ( x )$ is

A) $2 x \sin ^ { - 1 } x - \frac { x ^ { 2 } } { \sqrt { 1 - x ^ { 2 } } }$
B) $2 x \sin ^ { - 1 } x + \frac { x ^ { 2 } } { \sqrt { 1 - x ^ { 2 } } }$
C) $2 x \sin ^ { - 1 } x + \frac { x ^ { 2 } } { \sqrt { x ^ { 2 } - 1 } }$
D) $2 x \sin ^ { - 1 } x - \frac { x ^ { 2 } } { \sqrt { x ^ { 2 } - 1 } }$
E) $2 x \sin ^ { - 1 } x + \frac { x ^ { 2 } } { 1 + x ^ { 2 } }$

Question 32

Let $f ( x ) = \tan ^ { - 1 } ( \cos x )$ Then $f ^ { \prime } ( x )$ is

A) $\frac { \sin x } { 1 + \cos ^ { 2 } x }$
B) $- \frac { \sin x } { 1 + \cos ^ { 2 } x }$
C) $\frac { \sin x } { \cos ^ { 2 } x - 1 }$
D) $\frac { \sin x } { \cos ^ { 2 } x - 1 }$ .
E) $- \frac { \sin x } { \cos ^ { 2 } x - 1 }$

Question 33

Let $f ( x ) = \tan ^ { - 1 } ( \sqrt { x } )$ Then $f ^ { \prime } ( x )$ is

A) $\frac { 1 } { 2 \sqrt { x } \left( 1 + x ^ { 2 } \right) }$
B) $\frac { 2 } { \sqrt { x } \left( 1 + x ^ { 2 } \right) }$
C) $\frac { 1 } { 2 \sqrt { x } \left( 1 - x ^ { 2 } \right) }$
D) $\frac { 1 } { 2 \sqrt { x } ( 1 + x ) }$
E) $\frac { 1 } { 2 \sqrt { x } ( 1 - x ) }$

Question 34

Let $f ( x ) = \sec ^ { - 1 } \left( e ^ { x } \right)$ Then $f ^ { \prime } ( x )$ is

A) $\frac { 1 } { \sqrt { e ^ { 2 x } - 1 } }$
B) $- \frac { 1 } { e ^ { x } \sqrt { e ^ { 2 x } - 1 } }$
C) $\frac { 1 } { e ^ { x } \left( e ^ { 2 x } + 1 \right) }$
D) $- \frac { 1 } { e ^ { x } \left( e ^ { 2 x } + 1 \right) }$
E) $\frac { e ^ { x } } { \sqrt { e ^ { 2 x } - 1 } }$

Question 35

Let $f ( x ) = \tan ^ { - 1 } \left( 3 ^ { x } \right)$ Then $f ^ { \prime } ( x )$ is

A) $\frac { 3 ^ { x } } { 1 + 3 ^ { 2 x } }$
B) $\frac { 3 ^ { x } } { 1 + 3 ^ { 2 x } }$ .
C) $\frac { 3 ^ { x } \ln 3 } { 1 + 3 ^ { 2 x } }$
D) $\frac { 3 ^ { x } \ln 3 } { \sqrt { 3 ^ { 2 x } - 1 } }$
E) $\frac { 3 ^ { x } \ln 3 } { \sqrt { 1 - 3 ^ { 2 x } } }$

Question 36

Let $f ( x ) = \cos ^ { - 1 } \left( x ^ { 4 } \right)$ Then $f ^ { \prime } ( x )$ is

A) $- \frac { 4 x ^ { 3 } } { \sqrt { 1 - x ^ { 2 } } }$
B) $- \frac { 4 x ^ { 3 } } { \sqrt { 1 - x ^ { 4 } } }$
C) $- \frac { 4 x ^ { 3 } } { \sqrt { 1 - x ^ { 6 } } }$
D) $- \frac { 4 x ^ { 3 } } { \sqrt { 1 - x ^ { 8 } } }$
E) $\frac { 4 x ^ { 3 } } { \sqrt { 1 - x ^ { 8 } } }$

Question 37

Let $f ( x ) = \cot ^ { - 1 } \left( e ^ { 2 x } \right)$ Then $f ^ { \prime } ( x )$ is

A) $- \frac { 2 e ^ { 2 x } } { \sqrt { 1 - e ^ { 4 x } } }$
B) $- \frac { 2 e ^ { 2 x } } { 1 + e ^ { 4 x } }$
C) $- \frac { 2 e ^ { 2 x } } { 1 + e ^ { 2 x } }$
D) $- \frac { 2 e ^ { 2 x } } { 1 - e ^ { 2 x } }$
E) $- \frac { 2 e ^ { 2 x } } { \sqrt { e ^ { 4 x } - 1 } }$

Question 38

Let $f ( x ) = \tan ^ { - 1 } ( \cot x )$ Then $f ^ { \prime } ( x )$ is

A)-1
B) $\frac { \sec ^ { 2 } x } { 1 + \cot ^ { 2 } x }$
C) $- \frac { \sec ^ { 2 } x } { 1 + \cot ^ { 2 } x }$
D)1
E) $\frac { \cos ^ { 2 } x } { 1 + \cot ^ { 2 } x }$

Question 39

Let $f ( x ) = \sin ^ { - 1 } ( \cos x )$ on the interval $( 0 , \pi )$ Then $f ^ { \prime } ( x )$ is

A)-1
B) $\frac { - \sin x } { 1 + \cos ^ { 2 } x }$
C) $- \frac { \sin x } { \sqrt { 1 + \cos ^ { 2 } x } }$
D)1
E) $\frac { \sin x } { \sqrt { 1 + \cos ^ { 2 } x } }$

Question 40

Let $f ( x ) = \cos ^ { - 1 } ( \sin x )$ Then $f ^ { \prime } ( x )$ is

A)-1
B) $- \frac { \operatorname { cox } x } { 1 + \sin ^ { 2 } x }$
C) $- \frac { \operatorname { cox } x } { \sqrt { 1 + \sin ^ { 2 } x } }$
D)1
E) $\frac { \cos x } { \sqrt { 1 + \sin ^ { 2 } x } }$

Question 41

Let $f ( x ) = \ln \left( \frac { x + 3 } { x - 2 } \right)$ Then $f ^ { \prime } ( x )$ is

A) $- \frac { 1 } { ( x + 3 ) ( x - 2 ) }$
B) $- \frac { 5 } { ( x + 3 ) ( x - 2 ) }$
C) $\frac { 5 } { ( x + 3 ) ( x - 2 ) }$
D) $\frac { 1 } { ( x + 3 ) ( x - 2 ) }$
E) $- \frac { x - 2 } { 5 ( x + 3 ) }$

Question 42

Let $f ( x ) = \ln ( \sec x )$ Then $f ^ { \prime } ( x )$ is

A)sec x
B)tan² x
C)tan x
D)cot x
E)csc x

Question 43

Let $f ( x ) = \log _ { 2 } \sqrt { x ^ { 2 } + 5 }$ Then $f ^ { \prime } ( x )$ is

A) $\frac { 2 x } { \left( x ^ { 2 } + 5 \right) \ln 2 }$
B) $- \frac { 2 x } { \left( x ^ { 2 } + 5 \right) \ln 2 }$
C) $\frac { x } { 2 \left( x ^ { 2 } + 5 \right) }$
D) $\frac { x } { \left( x ^ { 2 } + 5 \right) \ln 2 }$
E) $- \frac { x } { \left( x ^ { 2 } + 5 \right) \ln 2 }$

Question 44

Let $f ( x ) = \tan ^ { - 1 } \left( \log _ { 3 } x \right)$ Then $f ^ { \prime } ( x )$ is

A) $- \frac { 1 } { x \left( 1 + \left( \log _ { 3 } x \right) ^ { 2 } \right) }$
B) $\frac { 1 } { x \left( 1 + \left( \log _ { 3 } x \right) ^ { 2 } \right) }$
C) $\frac { \ln 3 } { x \left( 1 + \left( \log _ { 3 } x \right) ^ { 2 } \right) }$
D) $- \frac { 1 } { x \ln 3 \left( 1 + \left( \log _ { 3 } x \right) ^ { 2 } \right) }$
E) $\frac { 1 } { x \ln 3 \left( 1 + \left( \log _ { 3 } x \right) ^ { 2 } \right) }$

Question 45

Let $f ( x ) = \ln \sqrt { \sin ( 2 x ) }$ Then $f ^ { \prime } ( x )$ is

A) $\sec ( 2 x )$
B) $\tan ^ { 2 } ( 2 x )$
C) $\tan ( 2 x )$
D) $\cot ( 2 x )$
E) $\csc ( 2 x )$

Question 46

Let $y = x ^ { x }$ Then $y ^ { \prime }$ is

A) $x ^ { x } ( \ln x - 1 )$
B) $x ^ { x } \ln x$
C) $x ^ { x } ( 1 + \ln x )$
D) $- x ^ { x } ( 1 + \ln x )$
E) $x ^ { x } ( 1 - \ln x )$

Question 47

Let $y = x ^ { x ^ { 2 } }$ Then $y ^ { \prime }$ is

A) $x ^ { x ^ { 2 } + 1 } ( 2 \ln x - 1 )$
B) $x ^ { x ^ { 2 } - 1 } ( 1 - 2 \ln x )$
C) $x ^ { x ^ { 2 } - 1 } ( 1 + 2 \ln x )$
D) $x ^ { x ^ { 2 } + 1 } ( 1 - 2 \ln x )$
E) $x ^ { x ^ { 2 } + 1 } ( 1 + 2 \ln x )$

Question 48

Let $f ( x ) = \ln ( \tan x )$ Then $f ^ { \prime } ( x )$ is

A) $\sin x \cos x$
B) $\frac { 1 } { \sin x \cos x }$
C) $- \frac { 1 } { \sin x \cos x }$
D)cot x
E)csc x

Question 49

Let $f ( x ) = \ln ( \cos x )$ Then $f ^ { \prime } ( x )$ is

A) $- \sin x \cos x$
B) $- \frac { 1 } { \sin x \cos x }$
C)-tan x
D)-cot x
E)-csc x

Question 50

Let $f ( x ) = \ln \left( \log _ { 2 } x \right)$ Then $f ^ { \prime } ( x )$ is

A) $\frac { 1 } { x \log _ { 2 } x }$
B) $\frac { \ln 2 } { x \log _ { 2 } x }$
C) $\frac { 1 } { x \ln x }$
D) $- \frac { 1 } { x \ln 2 ( \ln x ) }$
E) $\frac { \ln 2 } { x \ln x }$

Question 51

Let $y = x ^ { \ell ^ { x ^ { 2 } } }$ Then $y ^ { \prime }$ is

A) $e ^ { x } x ^ { e ^ { x } } ( 1 + x \ln x )$
B) $e ^ { x } x ^ { e ^ { x } } ( 1 - x \ln x )$
C) $e ^ { x } x ^ { e ^ { x } - 1 } ( 1 + x \ln x )$
D) $e ^ { x } x ^ { e ^ { x } - 1 } ( 1 - x \ln x )$
E) $e ^ { x } x ^ { e ^ { x } - 1 } ( 1 + \ln x )$

Question 52

Let $y = x ^ {e ^ { x}}$ Then $y ^ { \prime }$ is

A) $- \frac { x ^ { 2 } } { e ^ { \frac { 1 } { x } } }$
B) $- \frac { e ^ { \frac { 1 } { x } } } { x ^ { 2 } }$
C) $\frac { e ^ { \frac { 1 } { x } } } { x ^ { 2 } }$
D) $\frac { x ^ { 2 } } { e ^ { \frac { 1 } { x } } }$
E) $- \frac { x ^ { 2 } } { e ^ { x } }$

Question 53

Let $x ^ { y } = e \text {. }$ Then $y ^ { \prime }$ is

A) $- \frac { x } { ( \ln x ) ^ { 2 } }$
B) $\frac { x } { ( \ln x ) ^ { 2 } }$
C) $\frac { 1 } { x ( \ln x ) ^ { 2 } }$
D) $- \frac { 1 } { x ( \ln x ) ^ { 2 } }$
E) $\frac { x ^ { 2 } } { ( \ln x ) ^ { 2 } }$

Question 54

Let $y = \sin ( \ln x )$ Then $y ^ { \prime }$ is

A) $- \frac { \cos ( \ln x ) } { x }$
B) $\frac { \cos ( \ln x ) } { x }$
C) $\frac { x } { \cos ( \ln x ) }$
D) $- \frac { x } { \cos ( \ln x ) }$
E) $\frac { x } { \sin ( \ln x ) }$

Question 55

Let $y = \ln \left( \tan ^ { - 1 } x \right)$ Then $y ^ { \prime }$ is

A) $- \frac { \tan ^ { - 1 } x } { x ^ { 2 } + 1 }$
B) $\frac { \tan ^ { - 1 } x } { x ^ { 2 } + 1 }$
C) $\frac { x ^ { 2 } + 1 } { \tan ^ { - 1 } x }$
D) $\frac { 1 } { \tan ^ { - 1 } x \left( x ^ { 2 } + 1 \right) }$
E) $- \frac { 1 } { \tan ^ { - 1 } x \left( x ^ { 2 } + 1 \right) }$

Question 56

Let $y = \log _ { 3 } ( \cot x )$ Then $y ^ { \prime }$ is

A) $\frac { 1 } { \sin x \cos x \ln 3 }$
B) $- \frac { 1 } { \sin x \cos x \ln 3 }$
C) $\frac { \ln 3 } { \sin x \cos x }$
D) $\frac { 1 } { \sin x \cos x }$
E) $- \frac { \ln 3 } { \sin x \cos x }$

Question 57

Let $y = \ln ( x y )$ Using implicit differentiation, $y ^ { \prime }$ is

A) $\frac { x } { y ( y - 1 ) }$
B) $\frac { x } { y ( 1 - y ) }$
C) $\frac { y } { x ( y - 1 ) }$
D) $\frac { y } { x ( 1 - y ) }$
E) $\frac { x } { y ( x - 1 ) }$

Question 58

Let $y = \ln \left( \frac { x } { y } \right)$ Using implicit differentiation, $y ^ { \prime }$ is

A) $\frac { y } { x ( 1 + y ) }$
B) $\frac { y } { x ( 1 - y ) }$
C) $\frac { y } { x ( y - 1 ) }$
D) $\frac { x } { y ( 1 - y ) }$
E) $\frac { x } { y ( x - 1 ) }$

Question 59

Let $y = \ln \left( x ^ { y } \right)$ assuming $y \neq 0$ Using implicit differentiation, $y ^ { \prime }$ is

A) $- \frac { y } { x ( 1 + \ln x ) }$
B) $\frac { y } { x ( 1 + \ln x ) }$
C) $\frac { y } { x ( 1 - \ln x ) }$
D) $- \frac { y } { x ( 1 - \ln x ) }$
E) $\frac { y } { x ( 1 + \ln y ) }$

Question 60

Let $y = \ln \left( 2 x + y ^ { 2 } \right)$ Using implicit differentiation, $y ^ { \prime }$ is

A) $\frac { 2 } { 2 x - y ^ { 2 } - 2 y }$
B) $\frac { 2 } { 2 x + y ^ { 2 } + 2 y }$
C) $\frac { 2 } { 2 x - y ^ { 2 } + 2 y }$
D) $\frac { 2 } { 2 x + y ^ { 2 } - 2 y }$
E) $\frac { 2 } { - 2 x + y ^ { 2 } - 2 y }$

Question 61

Let $y = \sqrt { x }$ Then dy is

A) $- \frac { d x } { 2 \sqrt { x } }$
B) $\frac { d x } { 2 \sqrt { x } }$
C) $\frac { \sqrt { x } d x } { 2 }$
D) $- \frac { \sqrt { x } d x } { 2 }$
E) $2 \sqrt { x } d x$

Question 62

Let $y = e ^ { \cos x }$ Then dy is

A) $- e ^ { \sin x } d x$
B) $e ^ { \sin x } d x$
C) $e ^ { \cos x } \sin x d x$
D) $- e ^ { \cos x } \sin x d x$
E) $- e ^ { \cos x } \sin x \ln x d x$

Question 63

Let $y = 2 ^ { \tan x }$ Then dy is

A) $2 ^ { \sec ^ { 2 } x } \ln 2 d x$
B) $2 ^ { \sec ^ { 2 } x } \tan x d x$
C) $2 ^ { \tan x } \sec ^ { 2 } x d x$
D) $2 ^ { \tan x } \sec ^ { 2 } x \ln 2 d x$
E) $- 2 ^ { \tan x } \sec ^ { 2 } x \ln 2 d x$

Question 64

Let $y = \ln \left( 5 ^ { x } \right)$ Then dy is

A) $\frac { 1 } { \ln 5 } d x$
B) $\frac { 1 } { 5 ^ { x } } d x$
C) $5 ^ { x } \ln 5 d x$
D) $\frac { 5 ^ { x } } { \ln 5 } d x$
E) $\ln 5 d x$

Question 65

Let $y = \log _ { 2 } \left( \tan ^ { 3 } x \right)$ Then dy is

A) $\frac { 3 \sec x \sin x } { \ln 2 } d x$
B) $\frac { 3 \cos x \csc x } { \ln 2 } d x$
C) $\frac { 3 \sec x \csc x } { \ln 2 } d x$
D) $\frac { 3 \cos x \sin x } { \ln 2 } d x$
E) $\frac { \cos x \sin x } { 3 \ln 2 } d x$

Question 66

Let $y = \sin ( \cos x )$ Then dy is

A) $\cos ( \sin x ) d x$
B) $\cos ( - \sin x ) d x$
C) $\sin x [ \cos ( \cos x ) ] d x$
D) $- \sin x [ \cos ( \cos x ) ] d x$
E) $- \sin x [ \cos ( \sin x ) ] d x$

Question 67

Let $y = \tan ^ { - 1 } ( \ln x )$ Then dy is

A) $\frac { 1 } { 1 + ( \ln x ) ^ { 2 } } d x$
B) $- \frac { x } { 1 + ( \ln x ) ^ { 2 } } d x$
C) $\frac { x } { 1 + ( \ln x ) ^ { 2 } } d x$
D) $- \frac { 1 } { x \left[ 1 + ( \ln x ) ^ { 2 } \right] } d x$
E) $\frac { 1 } { x \left[ 1 + ( \ln x ) ^ { 2 } \right] } d x$

Question 68

Let $y = \frac { x } { e ^ { x } } .$ Then dy is

A) $\frac { ( 1 - x ) } { e ^ { x } d x }$
B) $\frac { x - 1 } { e ^ { x } } d x$
C) $\frac { e ^ { x } } { 1 - x } d x$
D) $\frac { e ^ { x } } { x - 1 } d x$
E) $\frac { e ^ { x } } { 1 - e ^ { x } } d x$

Question 69

Let $y = \tan \left( e ^ { x } \right)$ . Then dy is

A) $\sec ^ { 2 } \left( e ^ { x } \right) d x$
B) $e ^ { x } \sec ^ { 2 } \left( e ^ { x } \right) d x$
C) $\frac { \sec ^ { 2 } \left( e ^ { x } \right) } { 1 + e ^ { 2 x } } d x$
D) $\frac { \sec ^ { 2 } \left( e ^ { x } \right) } { 1 - e ^ { 2 x } } d x$
E) $\frac { \sec ^ { 2 } \left( e ^ { x } \right) } { e ^ { 2 x } - 1 } d x$

Question 70

Let $y = \sin \left( 7 ^ { x } \right)$ Then dy is

A) $7 ^ { x } \cos \left( 7 ^ { x } \right) d x$
B) $\cos \left( 7 ^ { x } \right) \ln 7 d x$
C) $7 ^ { x } \cos \left( 7 ^ { x } \right) \ln 7 d x$
D) $- 7 ^ { x } \cos \left( 7 ^ { x } \right) \ln 7 d x$
E) $- \cos \left( 7 ^ { x } \right) \ln 7 d x$

Question 71

The linear approximation $L ( x )$ to $f ( x ) = \sqrt { x }$ near $x _ { 0 } = \frac { 1 } { 4 }$ is

A) $- \frac { 1 } { 2 } + \left( x + \frac { 1 } { 4 } \right)$
B) $- \frac { 1 } { 2 } - \left( x - \frac { 1 } { 4 } \right)$
C) $\frac { 1 } { 2 } - \left( x - \frac { 1 } { 4 } \right)$
D) $- \frac { 1 } { 2 } + \left( x - \frac { 1 } { 4 } \right)$
E) $\frac { 1 } { 2 } + \left( x - \frac { 1 } { 4 } \right)$

Question 72

The linear approximation $L ( x )$ to $f ( x ) = \ln x$ near $x _ { 0 } = e$ is

A) $e + \frac { 1 } { e } ( x - e )$
B) $1 + \frac { 1 } { e } ( x - e )$
C) $- 1 + \frac { 1 } { e } ( x - e )$
D) $e - \frac { 1 } { e } ( x - e )$
E) $1 - \frac { 1 } { e } ( x - e )$

Question 73

The linear approximation $L ( x )$ to $f ( x ) = e ^ { - x }$ near $x _ { 0 } = 1$ is

A) $\frac { 1 } { e ^ { ( x - 1 ) } }$
B) $\frac { 1 } { e } ( x - 2 )$
C) $\frac { 1 } { e } - ( x - 1 )$
D) $\frac { 1 } { e } ( 2 - x )$
E) $\frac { 1 } { e } + ( x - 1 )$

Question 74

The linear approximation $L ( x )$ to $f ( x ) = \sqrt [ 3 ] { x }$ near $x _ { 0 } = 8$ is

A) $- \frac { 1 } { 2 } + \frac { 1 } { 12 } ( x - 8 )$
B) $- 2 - \frac { 1 } { 12 } ( x - 8 )$
C) $2 + \frac { 1 } { 12 } ( x - 8 )$
D) $2 - \frac { 1 } { 12 } ( x - 8 )$
E) $- 2 + \frac { 1 } { 12 } ( x - 8 )$

Question 75

The linear approximation $L ( x )$ to $f ( x ) = \tan x$ near $x _ { 0 } = \frac { \pi } { 4 }$ is

A) $1 + 2 \left( x - \frac { \pi } { 4 } \right)$
B) $1 - 2 \left( x - \frac { \pi } { 4 } \right)$
C) $1 + \left( x - \frac { \pi } { 4 } \right)$
D) $1 - \left( x - \frac { \pi } { 4 } \right)$
E) $- 1 + 2 \left( x - \frac { \pi } { 4 } \right)$

Question 76

The linear approximation $L ( x )$ to $f ( x ) = \sin x$ near $x _ { 0 } = - \frac { \pi } { 6 }$ is

A) $- \frac { 1 } { 2 } - \frac { \sqrt { 3 } } { 2 } \left( x + \frac { \pi } { 6 } \right)$
B) $- \frac { 1 } { 2 } + \frac { \sqrt { 3 } } { 2 } \left( x + \frac { \pi } { 6 } \right)$
C) $- \frac { \sqrt { 3 } } { 2 } + \frac { 1 } { 2 } \left( x + \frac { \pi } { 6 } \right)$
D) $- \frac { \sqrt { 3 } } { 2 } - \frac { 1 } { 2 } \left( x + \frac { \pi } { 6 } \right)$
E) $- \frac { 1 } { 2 } + \frac { \sqrt { 3 } } { 2 } \left( x - \frac { \pi } { 6 } \right)$

Question 77

Letting $f ( x ) = \sqrt { x }$ and $x _ { 0 } = 25$ the approximation of $\sqrt { 24 }$ by differentials is

A)5.1
B)5.05
C)4.95
D)4.9
E)4.85

Question 78

Letting $f ( x ) = \sqrt { x }$ and $x _ { 0 } = 25 ,$ the approximation of $\sqrt { 25.1 }$ by differentials is

A)5.1
B)5.05
C)5.01
D)4.99
E)4.95

Question 79

Letting $f ( x ) = \sqrt [ 3 ] { x }$ and $x _ { 0 } = 216$ the approximation of $\sqrt { 215 }$ by differentials is

A)5.94
B)5.97
C)5.99
D)6.01
E)6.05

Question 80

Letting $f ( x ) = \tan x$ and $x _ { 0 } = \frac { \pi } { 4 }$ the approximation of tan 44° by differentials is

A)0.962
B)0.963
C)0.964
D)0.965
E)0.966