Deck 3: The Derivative

Question 1

The slope of the tangent line $f ( x ) = - \frac { 1 } { 2 } x ^ { 2 }$ to the graph at the point (2,-2) is

A)-2
B)2
C) $- \frac { 1 } { 2 }$
D) $\frac { 1 } { 2 }$
E)-4

Answer 1

-2

Question 2

The slope of the tangent line to the graph $f ( x ) = 3 \sqrt { x }$ at the point (4,6) is

A) $- \frac { 3 } { 4 }$
B) $\frac { 3 } { 4 }$
C) $- \frac { 4 } { 3 }$
D) $\frac { 4 } { 3 }$
E) $\frac { 3 } { 2 }$

Answer 2

$\frac { 3 } { 4 }$

Question 3

The slope of the tangent line to the graph $f ( x ) = \frac { 1 } { x ^ { 2 } }$ at the point (-1,1) is

A)-3
B)-2
C)-1
D)1
E)2

Answer 3

2

Question 4

The slope of the tangent line to the graph $f ( x ) = \frac { 2 } { x + 4 }$ at the point (-3, 2) is

A)-3
B)-2
C)2
D)3
E)4

Question 5

An equation of the tangent line to the graph $f ( x ) = x ^ { 2 } - x$ at the point (-2, 6) is

A) $y = - 5 x - 4$
B) $y = - 5 x - 2$
C) $y = - 5 x + 2$
D) $y = - 5 x + 3$
E) $y = - 5 x + 4$

Question 6

An equation of the tangent line to the graph $f ( x ) = 2 x - x ^ { 2 }$ at the point (-1,-3) is

A) $y = 4 x - 3$
B) $y = 4 x - 1$
C) $y = 4 x + 1$
D) $y = 4 x + 3$
E) $y = 4 x + 5$

Question 7

An equation of the tangent line to the graph $f ( x ) = \frac { 3 } { 2 x - 1 }$ at the point (1, 3) is

A) $y = - 6 x + 9$
B) $y = - 6 x + 3$
C) $y = - 6 x + 1$
D) $y = - 6 x - 1$
E) $y = - 6 x - 3$

Question 8

An equation of the tangent line to the graph $f ( x ) = \sqrt { x + 3 }$ at the point (6, 3) is

A) $y = \frac { 1 } { 6 } x + 3$
B) $y = \frac { 1 } { 6 } x + 6$
C) $y = \frac { 1 } { 6 } x + 2$
D) $y = \frac { 1 } { 6 } x + \frac { 7 } { 3 }$
E) $y = \frac { 1 } { 6 } x + \frac { 5 } { 3 }$

Question 9

The instantaneous rate of change of $f ( x ) = 4 x - 3$ at c = 5 is

A)-3
B)4
C)5
D)17
E)20

Question 10

The instantaneous rate of change of $f ( x ) = \frac { 8 } { x + 1 }$ at c = 3 is

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

Question 11

The instantaneous rate of change of $f ( x ) = 3 \sqrt { x + 5 }$ at c = 4 is

A) $- \frac { 3 } { 2 }$
B) $- \frac { 1 } { 2 }$
C) $\frac { 1 } { 2 }$
D)1
E) $\frac { 3 } { 2 }$

Question 12

The instantaneous rate of change of $f ( x ) = 3 x - 2 x ^ { 2 }$ at c = 2 is

A)-5
B)-3
C)-2
D)3
E)5

Question 13

The instantaneous rate of change of $f ( x ) = \frac { 3 } { x + 1 }$ at c = 2 is

A) -3

B)
$- \frac { 1 } { 3 }$
C)
$\frac { 1 } { 3 }$
D) 1

E) 3

Question 14

The instantaneous rate of change of $f ( x ) = \sqrt { x } + 5$ at c = 9 is

A) $- \frac { 1 } { 6 }$
B) $- \frac { 1 } { 3 }$
C) $- \frac { 1 } { 2 }$
D) $\frac { 1 } { 6 }$
E) $\frac { 1 } { 3 }$

Question 15

The derivative of $f ( x ) = 3 x ^ { 2 } - 1$ at x = -1 is

A)-6
B)-3
C)-1
D)3
E)6

Question 16

The derivative of $f ( x ) = \frac { 4 } { x } + 5$ at x = -2 is

A)-4
B)-2
C)-1
D)2
E)4

Question 17

The derivative of $f ( x ) = \sqrt { x - 4 }$ at x = 8 is

A) $- \frac { 1 } { 8 }$
B) $- \frac { 1 } { 4 }$
C) $- \frac { 1 } { 2 }$
D) $\frac { 1 } { 4 }$
E) $\frac { 1 } { 8 }$

Question 18

The derivative of $f ( x ) = - 8 x + 7$ at x = 0 is

A)-8
B)7
C)0
D)8
E)-7

Question 19

The derivative of $f ( x ) = \frac { 4 } { x + 5 }$ at x = -4 is

A)-5
B)-4
C)1
D)4
E)5

Question 20

The derivative of $f ( x ) = \sqrt { x } - 4$ at x = 4 is

A)-4
B) $- \frac { 1 } { 4 }$
C) $- \frac { 1 } { 2 }$
D) $\frac { 1 } { 4 }$
E) $\frac { 1 } { 2 }$

Question 21

A pump is used to empty a 350-gallon hot tub. The table shows the amount of water, W(t), measured in gallons, remaining in the hot tub at time t, measured in minutes after the pump is turned on. $$\begin{array} { l l l l l l l } T & 0 & 15 & 30 & 45 & 60 & 90 \\W ( t ) & 350 & 320 & 280 & 225 & 170 & 105\end{array}$$ Approximate W'(30) in this context.

A)Water is draining at a rate of 28/3 gallons per minute when the pump has been operating for 30 minutes.
B)Forty gallons of water have been removed from the hot tub in 15 minutes of time.
C)There are 280 gallons of water in the hot tub 30 minutes after the pump is turned on.
D)Water is draining from the hot tub at a rate of 8/3 gallons per minute when the pump has been operating for 30 minutes.
E)Water is draining from the hot tub at a rate of 280 gallons per half-hour.

Question 22

A dehumidifier is used to reduce the moisture in a basement. The level of moisture reported on the display panel when the dehumidifier is turned on is 70%. The percent moisture, P(t), at time t is recorded for different times t in the table. $$\begin{array} { l l l l l l l } T & 0 & 2 & 6 & 15 & 20 & 30 \\P ( t ) & 70 & 67 & 60 & 53 & 51 & 50\end{array}$$ Estimate P'(20).

A)-0.20% per minute
B)-0.10% per minute
C)0.10% per minute
D)0.20% per minute
E)0.40% per minute

Question 23

Let $f ( x ) = 4 - x ^ { 2 }$ . Then $f ^ { \prime } ( x )$ is

A)-2x
B) $- x ^ { 2 }$
C)2x
D) $x ^ { 2 }$
E) $4 - 2 x$

Question 24

Let $f ( x ) = \frac { 1 } { 2 } - 4 x$ Then $f ^ { \prime } ( x )$ is

A)-4
B) $- \frac { 7 } { 2 }$
C) $\frac { 1 } { 2 }$
D) $- \frac { 1 } { 2 }$
E)4

Question 25

Let $f ( x ) = \sqrt { 9 - x }$ Then $f ^ { \prime } ( x )$ is

A) $\frac { \sqrt { 9 - x } } { 2 }$
B) $\frac { 2 } { \sqrt { 9 - x } }$
C) $\frac { 1 } { 2 \sqrt { 9 - x } }$
D) $- \frac { 1 } { 2 \sqrt { 9 - x } }$
E) $- \frac { 2 } { \sqrt { 9 - x } }$

Question 26

Let $f ( x ) = \frac { 8 } { \sqrt { 4 - x } }$ Then $f ^ { \prime } ( x )$ is

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

Question 27

Let $f ( x ) = \frac { 3 } { \sqrt { x } } + 4$ Then $f ^ { \prime } ( x )$ is

A) $- \frac { 3 } { 2 x ^ { \frac { 3 } { 2 } } }$
B) $- \frac { 3 } { 2 x ^ { \frac { 3 } { 2 } } } + 4$
C) $\frac { 3 } { 2 x ^ { \frac { 3 } { 2 } } }$
D) $\frac { 3 } { x ^ { \frac { 3 } { 2 } } }$
E) $\frac { 3 } { 2 x ^ { \frac { 3 } { 2 } } } + 4$

Question 28

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

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

Question 29

Let $f ( x ) = \sqrt { 9 - x }$ Then $f ^ { \prime } ( x )$ does not exist if x is

A)-9
B)-5
C)5
D)6
E)9

Question 30

Let $f ( x ) = \frac { 3 } { \sqrt { x } } + 4$ . Then does not exist if x is

A)0
B)1
C)4
D)9
E)16

Question 31

Let $f ( x ) = x - \frac { 2 } { x }$ . Then $f ^ { \prime } ( x )$ does not exist if x is

A)-2
B)-1
C)0
D)1
E)2

Question 32

Let $f ( x ) = \frac { 8 } { \sqrt { 5 - x } }$ Then $f ^ { \prime } ( x )$ does not exist if x is

A)-5
B)1
C)2
D)3
E)5

Question 33

Let $f ( x ) = | x |$ Which of the following is true?

A) $f$ is discontinuous at 0 .
B) $f$ has a derivative at 0.
C) $f$ has a corner at (0,0).
D) $f$ has a has a horizontal tangent line at (0,0).
E) $f$ has a vertical tangent line at (0,0).

Question 34

Let $f ( x ) = \sqrt [ 3 ] { x }$ Which of the following is true?

A) $f$ is discontinuous at 0.
B) $f$ has a derivative at 0.
C) $f$ has a corner at (0,0).
D) $f$ has a has a horizontal tangent line at (0,0).
E) $f$ has a vertical tangent line at (0,0).

Question 35

Let $f ( x ) = \lfloor x \rfloor$ Which of the following is true?

A) $f$ is discontinuous at 0.
B) $f$ has a derivative at 0.
C) $f$ has a corner at (0,0).
D) $f$ has a has a horizontal tangent line at (0,0).
E) $f$ has a vertical tangent line at (0,0).

Question 36

Let $f ( x ) = - x ^ { 2 }$ Which of the following is true?

A) $f$ is discontinuous at 0.
B) $f$ does not have a derivative at 0.
C) $f$ has a corner at (0,0).
D) $f$ has a has a horizontal tangent line at (0,0).
E) $f$ has a vertical tangent line at (0,0).

Question 37

Let $f ( x ) = 2 x ^ { 3 }$ Which of the following is true?

A) $f$ is discontinuous at 0.
B) $f$ does not have a derivative at 0.
C) $f$ has a corner at (0,0).
D) $f$ has a has a horizontal tangent line at (0,0).
E) $f$ has a vertical tangent line at (0,0).

Question 38

Let $f ( x ) = \left\{ \begin{array} { l l } 2 x + 1 & x < 0 \\- x + 1 & x > 0\end{array} \right.$ Which of the following is true?

A) $f$ is discontinuous at 0.
B) $f$ has a derivative at 0.
C) $f$ has a corner at (0,0).
D) $f$ has a has a horizontal tangent line at (0,0).
E) $f$ has a vertical tangent line at (0,0).

Question 39

Let $f ( x ) = \left\{ \begin{array} { c c } x ^ { 2 } & x \leq 0 \\- x ^ { 2 } & x > 0\end{array} \right.$ Which of the following is true?

A) $f$ is discontinuous at 0.
B) $f$ does not have a derivative at 0.
C) $f$ has a corner at (0,0).
D) $f$ has a has a horizontal tangent line at (0,0).
E) $f$ has a vertical tangent line at (0,0).

Question 40

Let $f ( x ) = 3 x - 2$ Which of the following is true?

A) $f$ is discontinuous at 0.
B) $f$ has a derivative at 0.
C) $f$ has a corner at (0,0).
D) $f$ has a horizontal tangent line at (0,0).
E) $f$ has a vertical tangent line at (0,0).

Question 41

Let $f(x)=\lfloor x\rfloor$
Which of the following is true?

A) $f$ is discontinuous at 0.
B) $f$ has a derivative at 0.
C) $f$ has a corner at (0,0).
D) $f$ has a horizontal tangent line at (0,0).
E) $f$ has a vertical tangent line at (0,0).

Question 42

Let $f ( x ) = - 4 x + 2$ . Which of the following is true?

A) $f$ is discontinuous at 0.
B) $f$ has a derivative at 0.
C) $f$ has a corner at (0,0).
D) $f$ has a horizontal tangent line at (0,0).
E) $f$ has a vertical tangent line at (0,0).

Question 43

Let $f ( x ) = \frac { 1 } { e }$ Then $f ^ { \prime } ( x )$ is

A) $- \frac { 1 } { e ^ { 2 } }$
B) $\frac { 1 } { e ^ { 2 } }$
C) $- \frac { 2 } { e ^ { 2 } }$
D) $\frac { 2 } { e ^ { 2 } }$
E)0

Question 44

Let $f ( x ) = \frac { 5 } { \pi }$ Then $f ^ { \prime } ( x )$ is

A)0
B) $- \frac { 5 } { \pi ^ { 2 } }$
C) $- \frac { 5 } { \pi }$
D) $\frac { 5 } { \pi }$
E) $\frac { 5 } { \pi ^ { 2 } }$

Question 45

Let $f ( x ) = \frac { 1 } { 2 } x ^ { 2 } - 3 x + 4$ Then $f ^ { \prime } ( x ) = 0$ if x is

A)-3
B)-1
C)0
D)1
E)3

Question 46

Let $f ( x ) = \frac { x ^ { 3 } - 2 x ^ { 2 } } { x }$ Then $f ^ { \prime } ( x ) = 0$ if x is

A)-2
B)-1
C)0
D)1
E)2

Question 47

Let $f ( x ) = x - e ^ { x }$ Then $f ^ { \prime } ( x ) = 0$ if x is

A)-e
B)-1
C)0
D)1
E)e

Question 48

Let $f ( x ) = 6 x - e ^ { x }$ Then $f ^ { \prime } ( x ) = 0$ if x is

A)-6
B)-ln 6
C)0
D)ln 6
E)6

Question 49

Let $f ( x ) = \frac { 1 } { 2 } x ^ { 2 } - 3 x + 4$ Then $f ^ { \prime } ( x ) = 0$ for all x in

A) $( - \infty , \infty )$
B) $( - \infty , - 3 )$
C) $( - 3 , \infty )$
D) $( - 3,3 )$
E) $( 3 , \infty )$

Question 50

Let $f ( x ) = \frac { x ^ { 3 } - 2 x ^ { 2 } } { x }$ Then $f ^ { \prime } ( x ) = 0$ for all x in

A) $( - \infty , \infty )$
B) $( - \infty , 0 )$
C) $( - 1 , \infty )$
D) $( - \infty , 0 ) \cup ( 1 , \infty )$
E) $( 1 , \infty )$

Question 51

Let $f ( x ) = x - e ^ { x }$ Then $f ^ { \prime } ( x ) = 0$ for all x in

A) $( - \infty , \infty )$
B) $( - \infty , 0 )$
C) $( 0 , \infty )$
D) $( - \infty , e )$
E) $\left( - \infty , e ^ { 2 } \right)$

Question 52

Let $f ( x ) = 6 x - e ^ { x }$ Then $f ^ { \prime } ( x ) = 0$ for all x in

A) $( - \infty , \infty )$
B) $( - \infty , \ln 6 )$
C) $( \ln 6 , \infty )$
D) $( - \infty , 6 )$
E) $( - \infty , - \ln 6 )$

Question 53

Let $f ( x ) = 6 x - e ^ { x }$ Then which of the following is the largest interval, of those listed, for which $f ^ { \prime } ( x ) > 0$ for all x in that interval?

A) $( - \infty , \infty )$
B) $( - \infty , \ln 6 )$
C) $( \ln 6 , \infty )$
D) $( - \infty , - \ln 6 )$
E) $( - \infty , - \ln 6 )$ .

Question 54

Let $f ( x ) = x ^ { 3 } - 6 x ^ { 2 }$ Then which of the following is the largest interval, of those listed, for which $f ^ { \prime } ( x ) > 0$ for all x in that interval?

A) $( 4 , \infty )$
B) $( - \infty , 0 )$
C) $( 0,4 )$
D) $( - \infty , 0 ) \cup ( 4 , \infty )$
E) $( - \infty , \infty )$

Question 55

Let $f ( x ) = \frac { 1 } { 2 } x ^ { 2 } - 3 x + 4$ Then which of the following is the largest interval, of those listed, for which $f ^ { \prime } ( x ) > 0$ for all x in that interval?

A) $( - \infty , \infty )$
B) $( - \infty , - 3 )$
C) $( - \infty , 3 )$
D)(-3,3)
E) $( 3 , \infty )$

Question 56

Let $f ( x ) = x - e ^ { x }$ . Then $f ^ { \prime } ( x ) < 0$ for all x in

A) $( - \infty , \infty )$
B) $( - \infty , 0 )$
C) $( 0 , \infty )$
D) $( - \infty , e )$
E) $\left( - \infty , e ^ { 2 } \right)$

Question 57

Let $f ( x ) = 6 x - e ^ { x }$ . Then which of the following is the largest interval, of those listed, for which $f ^ { \prime } ( x ) < 0$ for all x in that interval?

A) $( - \infty , \infty )$
B) $( - \infty , \ln 6 )$
C) $( \ln 6 , \infty )$
D) $( - \infty , 6 )$
E) $( - \infty , - \ln 6 )$

Question 58

Let $f ( x ) = x + e ^ { x }$ Then which of the following is the largest interval, of those listed, for which $f ^ { \prime } ( x ) < 0$ for all x in that interval?

A) $( - \infty , \infty )$
B) $( - \infty , 0 )$
C) $( 0 , \infty )$
D) $( - \infty , 3 )$
E) $\varnothing$

Question 59

Let $f ( x ) = e ^ { x }$ Then which of the following is the largest interval, of those listed, for which $f ^ { \prime } ( x ) < 0$ for all x in that interval?

A) $( - \infty , \infty )$
B) $( - \infty , 0 )$
C) $( 0 , \infty )$
D) $( - \infty , 3 )$
E) $\varnothing$

Question 60

Let $f ( x ) = x - 4 e ^ { x }$ Then which of the following is the largest interval, of those listed, for which $f ^ { \prime } ( x ) < 0$ for all x in that interval?

A) $( - \infty , \infty )$
B) $( - \infty , - \ln 4 )$
C) $( - \ln 4 , \ln 4 )$
D) $( - \ln 4 , \infty )$
E) $\varnothing$

Question 61

Let $f ( x ) = 4 e ^ { x } + 3 x$ Then which of the following is the largest interval, of those listed, for which for all x in that interval?

A) $( - \infty , \infty )$
B) $\left( - \infty , - \ln \frac { 3 } { 4 } \right)$
C) $\left( - \ln \frac { 3 } { 4 } , \ln \frac { 3 } { 4 } \right)$
D) $\left( - \infty , \ln \frac { 3 } { 4 } \right)$
E) $\varnothing$

Question 62

Let $f ( x ) = x ^ { 3 } - 6 x ^ { 2 }$ Then which of the following is the largest interval, of those listed, for which $f ^ { \prime } ( x ) < 0$ for all x
In that interval?

A) $( 4 , \infty )$
B) $( - \infty , 0 )$
C)(0,4)
D) $( - \infty , 0 ) \cup ( 4 , \infty )$
E) $( - \infty , \infty )$

Question 63

What is the equation of the normal line to the graph of $f ( x ) = x ^ { 2 } - 2 x$ at $x = 1$ ?

A)y = x
B)y = -x
C)y + 1 = x - 1
D)x = 1
E)y = -1

Question 64

What is the equation of the normal line to the graph of $f ( x ) = 3 \sqrt { x } + 2$ at $x = 9$ ?

A) $y = - 2 x$
B) $y - 11 = - 2 ( x - 9 )$
C) $y - 11 = 0.5 ( x - 9 )$
D) $y = 0.5 x$
E) $y = 0.5 x - 9$

Question 65

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

A) $2 x - 4$
B) $2 x + 4$
C) $- 2 x - 4$
D) $- 2 x + 4$
E) $2 x + 8$

Question 66

Let $f ( x ) = - 3 ( x + 5 ) ^ { 2 }$ . Then $f ^ { \prime } ( x )$ is

A) $- 6 x$
B) $- 6 x - 15$
C) $- 6 x - 30$
D) $- 6 x + 15$
E) $- 6 x + 30$

Question 67

Let $f ( x ) = ( - 3 x + 5 ) ^ { 2 }$ Then $f ^ { \prime } ( x )$ is

A) $- 6 x$
B) $- 6 x - 10$
C) $18 x - 30$
D) $18 x + 30$
E) $18 x - 5$

Question 68

Let $f ( x ) = 4 x ^ { 3 } e ^ { x }$ Then $f ^ { \prime } ( x )$ is

A) $12 x ^ { 2 } e ^ { x }$
B) $12 x ^ { 2 } + e ^ { x }$
C) $4 x ^ { 2 } ( x + 3 )$
D) $4 x ^ { 2 } e ^ { x } ( x + 3 )$
E) $4 x ^ { 3 } e ^ { x - 1 }$

Question 69

Let $e ^ { x } / x ^ { 4 }$ . Then $f ^ { \prime } ( x )$ is

A) $\frac { e ^ { x } } { x ^ { 4 } }$
B) $\frac { e ^ { x } ( x - 4 ) } { x ^ { 8 } }$
C) $\frac { e ^ { x } ( x - 4 ) } { x ^ { 7 } }$
D) $\frac { e ^ { x } ( x - 4 ) } { x ^ { 6 } }$
E) $\frac { e ^ { x } ( x - 4 ) } { x ^ { 5 } }$

Question 70

Let $f ( x ) = \frac { x ^ { 4 } } { e ^ { x } }$ Then $f ^ { \prime } ( x )$ is

A) $\frac { 4 x ^ { 3 } } { e ^ { x } }$
B) $\frac { x ^ { 4 } } { e ^ { x } }$
C) $\frac { x ^ { 3 } ( 4 - x ) } { e ^ { x } }$
D) $\frac { x ^ { 3 } ( 4 - x ) } { e ^ { 2 x } }$
E) $\frac { x ^ { 4 } - 3 } { e ^ { x } }$

Question 71

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

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

Question 72

Let $f ( x ) = \frac { 3 } { x ^ { 2 } + 1 }$ Then $f ^ { \prime } ( x )$ is

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

Question 73

Let $f ( x ) = x ^ { 4 } - \frac { 4 } { x ^ { 4 } }$ Then $f ^ { \prime } ( x )$ is

A) $4 x ^ { 3 } - \frac { 1 } { x ^ { 3 } }$
B) $4 x ^ { 3 } - \frac { 4 } { x ^ { 5 } }$
C) $4 x ^ { 3 } - \frac { 16 } { x ^ { 5 } }$
D) $4 x ^ { 3 } + \frac { 4 } { x ^ { 5 } }$
E) $4 x ^ { 3 } + \frac { 16 } { x ^ { 5 } }$

Question 74

Let $f ( x ) = \frac { 4 x } { 3 x + 5 }$ Then $f ^ { \prime } ( x )$ is

A) $\frac { 4 } { 3 }$
B) $\frac { 20 } { 3 x + 5 }$
C) $- \frac { 20 } { ( 3 x + 5 ) ^ { 2 } }$
D) $\frac { 20 } { ( 3 x + 5 ) ^ { 2 } }$
E) $\frac { 24 x - 5 } { ( 3 x + 5 ) ^ { 2 } }$

Question 75

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

A) $3 e ^ { x }$
B) $3 e ^ { x } + 3 x ^ { 2 } e ^ { x - 1 }$
C) $3 x e ^ { x - 1 }$
D) $3 e ^ { x } ( x + 2 )$
E) $3 e ^ { x } ( x + 1 )$

Question 76

Let $f ( x ) = \frac { e ^ { x } } { x }$ Assuming $x \neq 0 , f ^ { \prime } ( x )$ is

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

Question 77

Let $f ( x ) = e ^ { 2 } x ^ { 3 }$ Then $f ^ { \prime } ( x )$ is

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

Question 78

Let $f ( x ) = \frac { x ^ { 2 } + x } { x }$ Assuming $x \neq 0 , f ^ { \prime } ( x )$ is

A) $\frac { 1 } { 2 }$
B)0
C) $x + \frac { 1 } { 2 }$
D) $\frac { x } { 2 } + 1$
E) $\frac { x } { 2 } - 1$

Question 79

Let $f ( x ) = x ^ { 3 } - 4 x ^ { 2 } + x - 5$ Then $f ^ { \prime } ( x )$ is

A) $6 x - 8$
B) $6 x$
C) $3 x ^ { 2 } - 8 x$
D) $- 6 x$
E) $- 6$

Question 80

The fourth derivative $\frac { d ^ { 4 } } { d x ^ { 4 } } \left( x ^ { 2 } - e ^ { x } \right)$ is

A) $2 x - e ^ { x }$
B) $2 - e ^ { x }$
C) $2 + e ^ { x }$
D) $- e ^ { x }$
E) $e ^ { x }$