Deck 11: Techniques of Differentiation

Question 1

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ using implicit differentiation. $9 x + 10 y = 7$ ?

A) $- \frac { 9 } { 10 }$
B) $- 9$
C) $0$
D) $- \frac { 10 } { 9 }$
E) $- 10$

Answer 1

$- \frac { 9 } { 10 }$

Question 2

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ using implicit differentiation. $x ^ { 2 } - 6 y = 7$ ?

A) $7$
B) $- \frac { 1 } { 6 }$
C) $\frac { x } { 6 }$
D) $- 3$
E) $\frac { x } { 3 }$

Answer 2

$\frac { x } { 3 }$

Question 3

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ using implicit differentiation. $e ^ { x } y = 4$ ?

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

Answer 3

$- y$

Question 4

Use logarithmic differentiation to find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ . $y = \left( x ^ { 3 } + x \right) \sqrt { x ^ { 3 } + 6 }$ ?

A) $\left( \frac { 3 x ^ { 2 } + 1 } { x ^ { 3 } + x } + \frac { 3 x ^ { 2 } } { 2 x ^ { 3 } + 12 } \right)$
B) $\left( x ^ { 3 } + x \right) \sqrt { x ^ { 3 } + 6 } \left( \frac { 1 } { x ^ { 3 } + x } + \frac { 1 } { 2 \left( x ^ { 3 } + 6 \right) } \right)$
C) $\left( x ^ { 3 } + x \right) \sqrt { x ^ { 3 } + 6 } \left( \frac { 3 x ^ { 2 } + 1 } { x ^ { 3 } + x } + \frac { 3 x ^ { 2 } } { 2 \left( x ^ { 3 } + 6 \right) } \right)$
D) $\left( 3 x ^ { 2 } + 1 \right) \left( \frac { 3 x ^ { 2 } + 1 } { x } + \frac { 3 x ^ { 2 } } { 2 x ^ { 3 } + 12 } \right)$
E) $3 x ^ { 2 } \sqrt { x ^ { 3 } + 12 } \left( \frac { 3 x ^ { 2 } + 1 } { x ^ { 3 } + x } + \frac { x ^ { 2 } } { 2 x ^ { 3 } + 6 } \right)$

Question 5

Use logarithmic differentiation to find .
​

Question 6

Find $\frac { \mathrm { d } s } { \mathrm {~d} t }$ using implicit differentiation. $e ^ { s t } = s ^ { 9 }$

A) $9 - e ^ { s t }$
B) $\frac { e ^ { s t } } { 9 - e ^ { s t } }$
C) $\frac { s e ^ { s t } } { 9 s ^ { 8 } - t e ^ { s t } }$
D) $\frac { s ^ { 8 } } { 9 s ^ { 8 } - t e ^ { s t } }$
E) $\frac { s e ^ { s t } } { 9 s - t e ^ { s t } }$

Question 7

The number P of CDs the Snappy Hardware Co. can manufacture at its plant in one day is given by
​ ​
where x is the number of workers at the plant and y is the annual expenditure at the plant (in dollars). Compute at a production level of 23,000 CDs per day and . Round your answer to two decimal places.

Question 8

Find the equation of the tangent line for $( x y ) ^ { 2 } + ( x y ) - x = 10$ at the point $( - 10,0 )$ .

A) $y = - 10 x - 1$
B) $y = - \frac { 1 } { 10 } x + 1$
C) $y = - \frac { 1 } { 10 } x - 1$
D) $y = - 10 x - 2$
E) $y = - \frac { 1 } { 10 } x - 2$

Question 9

The number P of CDs the Snappy Hardware Co. can manufacture at its plant in one day is given by $P = x ^ { 0.2 } y ^ { 0.8 }$
Where x is the number of workers at the plant and y is the annual expenditure at the plant (in dollars). Compute $\frac { \mathrm { d } y } { \mathrm {~d} x }$ at a production level of 24,000 CDs per day and $x = 105$ . Round your answer to two decimal places.

A)-250.64
B) -234.31
C) -222.19
D) -255.85
E) 234.31

Question 10

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ using implicit differentiation. $\frac { e ^ { x } } { y ^ { 2 } } = 16 + e ^ { y }$

A) $\frac { y e ^ { x } } { 16 e ^ { x } + 3 y ^ { 2 } e ^ { y } }$
B) $\frac { 2 y e ^ { x } } { e ^ { x } + y e ^ { y } }$
C) $\frac { 2 e ^ { x } + y ^ { 3 } e ^ { y } } { y e ^ { x } }$
D) $\frac { y e ^ { x } } { 2 e ^ { x } + y ^ { 3 } e ^ { y } }$
E) $\frac { y } { 2 + y ^ { 3 } }$

Question 11

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ using implicit differentiation. $x e ^ { y } - y e ^ { x } = 12$

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

Question 12

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ using implicit differentiation. $\ln \left( 15 + e ^ { x y } \right) = y$

A) $x + y$
B) $\frac { y e ^ { x y } } { 15 + e ^ { x y } }$
C) $\frac { y } { 1 - x }$
D) $\frac { y e ^ { x y } } { 15 + e ^ { x y } ( 1 - x ) }$
E) $\frac { 1 } { 15 + e ^ { x y } ( 1 - x ) }$

Question 13

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ using implicit differentiation. $x ^ { 2 } + y ^ { 2 } = 4$

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

Question 14

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ using implicit differentiation. $y \ln x + y = 2$

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

Question 15

An employment research company estimates that the value of a recent MBA graduate to an accounting company is $V = 5 e ^ { 2 } + 3 g ^ { 3 }$
Where V is the value of the graduate, ?e is a number of years of prior business experience, and g is the graduate school grade point average. If
$V = 240$ , find $\frac { \mathrm { de } e } { \mathrm {~d} g }$ when $g = 1$ .

All the answers were rounded to the nearest hundredth.

A)0.13
B) -1.58
C) -2.79
D) -0.13
E) -0.25

Question 16

Use logarithmic differentiation to find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ . Do not simplify the result. $y = ( 5 x + 1 ) ( 7 x - 1 )$

A) $\frac { d y } { d x } = ( 5 x + 1 ) ( 7 x - 1 ) \left( \frac { 5 } { 5 x + 1 } + \frac { 7 } { 7 x - 1 } \right)$
B) $\frac { \mathrm { d } y } { \mathrm {~d} x } = \left( \frac { 5 } { 5 x + 1 } + \frac { 7 } { 7 x - 1 } \right)$
C) $\frac { \mathrm { d } y } { \mathrm {~d} x } = ( 5 x + 1 ) ( 7 x - 1 ) \left( \frac { 5 } { 5 x + 1 } + \frac { 7 } { 7 x - 1 } \right) ^ { 2 }$
D) $\frac { \mathrm { d } y } { \mathrm {~d} x } = ( 7 x - 1 ) \left( \frac { 5 } { 5 x + 1 } + \frac { 7 } { 7 x - 1 } \right)$
E) $\frac { \mathrm { d } y } { \mathrm {~d} x } = ( 5 x + 1 ) \left( \frac { 5 } { 5 x + 1 } + \frac { 7 } { 7 x - 1 } \right)$

Question 17

Find the derivative of the following function. $f ( x ) = \ln ( 9 x - 17 )$

A) $\frac { 9 } { 9 x - 17 }$
B) $\frac { 1 } { 9 x - 17 }$
C) $\frac { 17 } { 9 x - 17 }$
D) $\frac { 153 } { 9 x - 17 }$
E) none of these

Question 18

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ using implicit differentiation. $\frac { x y } { 2 } - y ^ { 2 } = 9$

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

Question 19

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ using implicit differentiation. $10 x + 11 y = x y$ ?

A) $\frac { 10 - y } { x - 11 }$
B) $\frac { 10 - x } { y - 11 }$
C) $11 - y$
D) $\frac { x - 11 } { 10 - y }$
E) $x - 10$

Question 20

Find $\frac { \mathrm { d } x } { \mathrm {~d} y }$ using implicit differentiation. $( x y ) ^ { 2 } + y ^ { 2 } = 5$

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

Question 21

Find the derivative of the following function.
​

Question 22

Find the derivative of the following function.
$f ( x ) = \log _ { 5 } 8 x$

A) $\frac { 1 } { x \ln 5 }$
B) $\frac { 5 } { x \ln 8 }$
C) $\frac { 8 } { x \ln 5 }$
D) $\frac { 1 } { 8 x \ln 5 }$
E) none of these

Question 23

If $17,000 is invested in a savings account yielding 3% per year, compounded semiannually, how fast is the balance growing after 3 years Round your answer to the nearest cent.

A)$691.89 per year
B) $387.46 per year
C) $774.92 per year
D) $996.33 per year
E) $553.52 per year

Question 24

Calculate the derivative of the function. $\left( 4 x ^ { 2 } + 2 x + 2 \right) ^ { - 4 }$

A) $g ^ { \prime } ( x ) = \left( - 16 x ^ { 2 } + 8 x + 8 \right) ^ { - 5 }$
B) $g ^ { \prime } ( x ) = - 4 ( 8 x + 2 ) \left( 4 x ^ { 2 } + 2 x + 2 \right) ^ { - 5 }$
C) $g ^ { \prime } ( x ) = - 32 \left( 4 x ^ { 2 } + 2 x + 2 \right) ^ { - 5 }$
D) $g ^ { \prime } ( x ) = - 4 ( 8 x + 2 ) \left( 4 x ^ { 2 } + 2 x + 2 \right)$
E) $g ^ { \prime } ( x ) = - 4 \left( 4 x ^ { 2 } + 2 x + 2 \right) ^ { - 5 }$

Question 25

Find the equation of the straight line, tangent to $y = e ^ { 7 x } \log _ { 5 } x$ at the point $( 1,0 )$ .

A) $y ( x ) = \frac { e ^ { 7 } } { \ln 5 } x + \frac { e ^ { 7 } } { \ln 5 }$
B) $y ( x ) = \frac { e ^ { 5 } } { \ln 7 } x - \frac { e ^ { 5 } } { \ln 7 }$
C) $y ( x ) = \frac { e ^ { 5 } } { \ln 7 } x + \frac { e ^ { 5 } } { \ln 7 }$
D) $y ( x ) = \frac { e ^ { 7 } } { \ln 5 } x - \frac { e ^ { 7 } } { \ln 5 }$
E) none of these

Question 26

Find the derivative of the function. $h ( x ) = \ln [ ( 2 x + 8 ) ( 3 x + 6 ) ]$

A) $\frac { 3 } { ( 2 x + 8 ) } - \frac { 2 } { ( 3 x + 6 ) }$
B) $\frac { 3 } { ( 2 x + 8 ) } + \frac { 2 } { ( 3 x + 6 ) }$
C) $\frac { 1 } { ( 2 x + 8 ) } + \frac { 1 } { ( 3 x + 6 ) }$
D) $\frac { 1 } { ( 2 x + 8 ) } - \frac { 1 } { ( 3 x + 6 ) }$
E) $\frac { 2 } { ( 2 x + 8 ) } + \frac { 3 } { ( 3 x + 6 ) }$

Question 27

The population of Upper Anchora was 1,000,000 at the start of 1996 and was doubling every 9 years. How fast was it growing per year at the start of 1996 ?
Round your answer to the nearest thousand.

A)154,000 people per year
B) 78,000 people per year
C) 77,000 people per year
D) 76,000 people per year
E) 67,000 people per year

Question 28

Find the derivative of the function. $g ( x ) = \ln | 8 x - 9 |$

A) $\frac { 8 } { | 8 x - 9 | }$
B) $\frac { 1 } { 8 x - 9 }$
C) $\frac { 1 } { - 8 x + 9 }$
D) $\frac { 8 } { 8 x - 9 }$
E) none of these

Question 29

Find the indicated derivative. The independent variable is a function of t. $y = x ^ { 0.7 } ( 1 + x ) ; \frac { \mathrm { d } y } { \mathrm {~d} t } =$

A) $\frac { \mathrm { d } y } { \mathrm {~d} t } = \left( 0.7 x ^ { - 0.3 } \right) \frac { \mathrm { d } x } { \mathrm {~d} t }$
B) $\frac { \mathrm { d } y } { \mathrm {~d} t } = \left( 0.7 x ^ { - 0.3 } + 2.7 x ^ { 0.7 } \right) \frac { \mathrm { d } x } { \mathrm {~d} t }$
C) $\frac { \mathrm { d } y } { \mathrm {~d} t } = \left( 0.7 x ^ { - 0.3 } + 1.7 x ^ { 0.7 } \right) \frac { \mathrm { d } x } { \mathrm {~d} t }$
D) $\frac { \mathrm { d } y } { \mathrm {~d} t } = \left( 1.7 x ^ { 0.7 } \right) \frac { \mathrm { d } x } { \mathrm {~d} t }$
E) $\frac { \mathrm { d } y } { \mathrm {~d} t } = \left( 0.7 x ^ { 0.7 } + 2.7 x ^ { 0.7 } \right) \frac { \mathrm { d } x } { \mathrm {~d} t }$

Question 30

Find the derivative of the function. $r ( x ) = \left( e ^ { - 6 x ^ { 6 } } \right) ^ { 8 }$

A) $- 288 \left( e ^ { - 6 x ^ { 6 } } \right) ^ { 7 } x ^ { 5 }$
B) $- 48 e ^ { 8 } x ^ { 6 }$
C) $48 e ^ { 7 } x ^ { 6 }$
D) $- 288 \left( e ^ { - 6 x ^ { 6 } } \right) ^ { 8 } x ^ { 5 }$
E) none of these

Question 31

Find the derivative of the function.
​
​

Question 32

Find the derivative of the function.
$r ( x ) = \left[ \ln \left( x ^ { 8 } \right) \right] ^ { 3 }$

A) $\frac { 24 \left[ \ln \left( x ^ { 7 } \right) \right] ^ { 2 } } { x ^ { 8 } }$
B) $\frac { 24 \left[ \ln \left( x ^ { 8 } \right) \right] ^ { 2 } } { x ^ { 8 } }$

C) $\frac { 24 \left[ \ln \left( x ^ { 8 } \right) \right] ^ { 2 } } { x }$
D) $\frac { 24 \left[ \ln \left( x ^ { 8 } \right) \right] ^ { 3 } } { x ^ { 8 } }$
E) none of these

Question 33

If $24,000 is invested in a savings account yielding 6% per year, compounded semiannually, how fast is the balance growing after 2 years

Please enter your answer as a number (in $ per year) without the units. Round your answer to two decimal places.

Question 34

Calculate the derivative of the function. $s ( x ) = \left( \frac { 8 x + 5 } { 2 x - 6 } \right) ^ { 6 }$

A) $s ^ { \prime } ( x ) = - 6 \left( \frac { 8 x + 5 } { 2 x - 6 } \right) ^ { 5 } \frac { 58 x } { ( 2 x - 6 ) ^ { 2 } }$
B) $s ^ { \prime } ( x ) = - 6 \left( \frac { 8 x + 5 } { 2 x - 6 } \right) ^ { 5 } \frac { 48 } { ( 2 x - 6 ) ^ { 2 } }$
C) $s ^ { \prime } ( x ) = \left( \frac { 8 x + 5 } { 2 x - 6 } \right) ^ { 5 } \frac { 58 } { ( 2 x - 6 ) ^ { 2 } }$
D) $s ^ { \prime } ( x ) = - 6 \left( \frac { 8 x + 5 } { 2 x - 6 } \right) ^ { 5 } \frac { 58 } { ( 2 x - 6 ) ^ { 2 } }$
E) $s ^ { \prime } ( x ) = 6 \left( \frac { 8 x + 5 } { 2 x - 6 } \right) ^ { 5 }$

Question 35

Find the derivative of the function.
$f ( x ) = \ln \mid \frac { ( 3 x + 5 ) ^ { 2 } } { ( 7 x + 5 ) ^ { 2 } ( 2 x + 8 ) }$

A) $\frac { 3 } { ( 3 x + 5 ) ^ { 2 } } - \frac { 7 } { ( 7 x + 5 ) ^ { 2 } } - \frac { 2 } { 2 x + 8 }$
B) $\frac { 6 } { 3 x + 5 } + \frac { 14 } { 7 x + 5 } + \frac { 2 } { 2 x + 8 }$
C) $\frac { 3 } { ( 3 x + 5 ) ^ { 2 } } + \frac { 7 } { ( 7 x + 5 ) ^ { 2 } } + \frac { 2 } { 2 x + 8 }$
D) $\frac { 6 } { 3 x + 5 } - \frac { 14 } { 7 x + 5 } - \frac { 2 } { 2 x + 8 }$
E) none of these

Question 36

Find the derivative of the function.
$h ( x ) = e ^ { 9 x ^ { 2 } - 4 x + \frac { 1 } { x } }$

A) $\frac { 9 x ^ { 3 } - 8 x ^ { 2 } - 1 } { x } e ^ { 9 x ^ { 2 } - 4 x + \frac { 1 } { x } }$
B) $\frac { 18 x ^ { 3 } - 4 x ^ { 2 } - 1 } { x ^ { 2 } } e ^ { 9 x ^ { 2 } - 4 x + \frac { 1 } { x } }$
C) $\frac { 9 x ^ { 3 } - 8 x ^ { 2 } - 1 } { x ^ { 2 } } e ^ { 9 x ^ { 2 } - 4 x + \frac { 1 } { x } }$
D) $\frac { 18 x ^ { 3 } - 4 x ^ { 2 } - 1 } { x } e ^ { 9 x ^ { 2 } - 4 x + \frac { 1 } { x } }$
E) none of these

Question 37

Find the derivative of the function. $\frac { e ^ { - 5 x } } { 5 x e ^ { 5 x } }$

A) $- \frac { 10 x - 1 } { 5 x ^ { 2 } e ^ { 10 x } }$
B) $\frac { 10 x + 1 } { 5 x ^ { 2 } e ^ { 10 x } }$
C) $- \frac { 10 x + 1 } { x ^ { 2 } e ^ { 10 x } }$
D) $- \frac { 10 x + 1 } { 5 x ^ { 2 } e ^ { 10 x } }$
E) none of these

Question 38

Find the derivative of the function.
$f ( x ) = e ^ { 4 x ^ { 6 } } \ln ( 8 x )$

A) $24 e ^ { 4 x ^ { 6 } } x ^ { 5 } \ln ( 8 x ) + \frac { e ^ { 4 x ^ { 6 } } } { x }$
B) $24 e ^ { 4 x ^ { 5 } } x ^ { 5 } \ln ( 8 x ) + \frac { e ^ { 4 x ^ { 6 } } } { x }$
C) $24 e ^ { 4 x ^ { 6 } } x ^ { 5 } \ln ( 8 x ) + \frac { e ^ { 4 x ^ { 6 } } } { 8 }$
D) $24 e ^ { 4 x ^ { 6 } } x ^ { 6 } \ln ( 8 x ) + \frac { 8 e ^ { 4 x ^ { 6 } } } { x }$
E) $6 e ^ { 4 x ^ { 6 } } x ^ { 5 } \ln ( 8 x ) + \frac { 8 e ^ { 4 x ^ { 6 } } } { x }$

Question 39

Find the derivative of the function. $r ( x ) = \ln \mid 4 x + e ^ { 4 x }$

A) $\frac { 4 + 4 e ^ { 4 x } } { 4 x + 4 e ^ { 4 x } }$
B) $\frac { 4 + 4 e ^ { 4 x } } { 4 x + e ^ { 4 x } }$
C) $\frac { 4 + e ^ { 4 x } } { 4 x + e ^ { 4 x } }$
D) $\frac { 4 + e ^ { 4 x } } { 4 x + 4 e ^ { 4 x } }$
E) none of these

Question 40

Find the derivative of the function.
$f ( x ) = \left( x ^ { 3 } + 5 \right) \ln x$

A) $\frac { x ^ { 3 } ( 1 + \ln x ) + 5 } { x }$
B) $\frac { x ^ { 3 } ( 1 + 3 \ln x ) + 5 } { x }$
C) $\frac { x ^ { 3 } ( 3 + 3 \ln x ) + 5 } { x }$
D) $\frac { x ^ { 2 } ( 1 + 3 \ln x ) + 5 } { x }$
E) none of these

Question 41

Find the derivative of the function. $f ( x ) = \frac { 4 } { x ^ { 3 } }$

A) $- \frac { 12 } { x ^ { 2 } }$
B) $12 x ^ { 2 }$
C) 4
D) $- \frac { 12 } { x ^ { 4 } }$
E) $- 12 x ^ { 2 }$

Question 42

Calculate $\frac { \mathrm { d } y } { \mathrm {~d} x }$ . You need not expand your answer. $y = \frac { 2 x + 2 } { 4 x - 5 }$

A) $\frac { 2 ( 4 x - 5 ) + 4 ( 2 x + 2 ) } { 4 x - 5 }$
B) $\frac { 2 ( 4 x - 5 ) - 4 ( 2 x + 2 ) } { ( 4 x - 5 ) ^ { 2 } }$
C) $\frac { 2 ( 4 x - 5 ) + 4 ( 2 x + 2 ) } { ( 4 x - 5 ) ^ { 2 } }$
D) $2 ( 4 x - 5 ) - 4 ( 2 x + 2 )$
E) 0.5

Question 43

Find the derivative of the function. $h ( x ) = x ( 4 + 4 x )$

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

Question 44

The demand for the Cyberpunk II arcade video game is modeled by the logistic curve $q ( t ) = \frac { 13,000 } { 1 + 0.6 e ^ { - 0.5 t } }$
Where $q ( t )$ is the total number of units sold t months after the game's introduction.

Use technology to estimate $q ^ { \prime } ( 9 )$ .

Assume that the manufacturers of Cyberpunk II sell each unit for $900. What is the company's marginal revenue, $\frac { \mathrm { d } R } { \mathrm {~d} q }$

Use the chain rule to estimate the rate at which revenue is growing 9 months after the introduction of the video game.

Please round each answer to the nearest whole number.

A) $\frac { \mathrm { d } q } { \mathrm {~d} t } = 43 , \frac { \mathrm { d } R } { \mathrm {~d} q } = 900 , \frac { \mathrm { d } R } { \mathrm {~d} t } = 38,734$
B) $\frac { \mathrm { d } q } { \mathrm {~d} t } = 43 , \frac { \mathrm { d } R } { \mathrm {~d} q } = 900 , \frac { \mathrm { d } R } { \mathrm {~d} t } = 38,478$
C) $\frac { \mathrm { d } q } { \mathrm {~d} t } = 71 , \frac { \mathrm { d } R } { \mathrm {~d} q } = 700 , \frac { \mathrm { d } R } { \mathrm {~d} t } = 64,130$
D) $\frac { \mathrm { d } q } { \mathrm {~d} t } = 86 , \frac { \mathrm { d } R } { \mathrm {~d} q } = 800 , \frac { \mathrm { d } R } { \mathrm {~d} t } = 76,956$
E) $\frac { \mathrm { d } q } { \mathrm {~d} t } = 143 , \frac { \mathrm { d } R } { \mathrm {~d} q } = 900 , \frac { \mathrm { d } R } { \mathrm {~d} t } = 128,260$

Question 45

The Pentagon is planning to build a new satellite that will be spherical. As is typical in these cases, the specifications keep changing, so that the size of the satellite keeps growing. In fact, the radius of the planned satellite is growing 0.9 foot/week. Its cost will be $1,400 per cubic foot. At the point when the plans call for a satellite 8 feet in radius, how fast is the cost growing (The volume of a solid sphere of radius r is $V = \frac { 4 } { 3 } \pi r ^ { 3 }$ .)

A) $\frac { \mathrm { d } P } { \mathrm {~d} t } = \frac { \$ 3,584,000 \pi } { \text { week } }$
B) $\frac { \mathrm { d } P } { \mathrm {~d} t } = \frac { \$ 40,320 \pi } { \text { week } }$
C) $\frac { \mathrm { d } P } { \mathrm {~d} t } = \frac { \$ 230 \pi } { \text { week } }$
D) $\frac { \mathrm { d } P } { \mathrm {~d} t } = \frac { \$ 161,280 \pi } { \text { week } }$
E) $\frac { \mathrm { d } P } { \mathrm {~d} t } = \frac { \$ 322,560 \pi } { \text { week } }$

Question 46

Compute the indicated derivative using the chain rule. $y = 7 x + 10 ; \frac { \mathrm { d } x } { \mathrm {~d} y }$

A) $\frac { 1 } { 10 }$
B) $- \frac { 7 } { 10 }$
C) $\frac { 1 } { 7 }$
D) 7
E) -10

Question 47

Calculate $\frac { \mathrm { d } y } { \mathrm {~d} x }$ . $y = x ^ { 2 } ( 3 x + 2 ) ( 5 x + 2 )$

A) $65 x ^ { 2 } + ( 3 x + 48 ) ( 5 x + 2 )$
B) $60 x ^ { 3 } + 48 x ^ { 2 } + 8 x$
C) $35 x ^ { 2 } + ( 3 x + 2 ) ( 5 x + 2 )$
D) $3 x ^ { 3 } + 48 x ^ { 2 } + 8 x$
E) $3 x ^ { 2 } + 48 x + 8$

Question 48

Compute the indicated derivative using the chain rule. $y = 7 x - 6$ ; $\frac { \mathrm { d } x } { \mathrm {~d} y }$

A) $\frac { 7 } { 6 }$
B) 6
C) 7
D) $\frac { 1 } { 6 }$
E) $\frac { 1 } { 7 }$

Question 49

Compute the indicated derivative using the chain rule. $y = 10 x ^ { 2 } - 7 x ; \left. \frac { \mathrm { d } x } { \mathrm {~d} y } \right| _ { x = 2 }$

A) $\frac { 1 } { 33 }$
B) $\frac { 7 } { 10 }$
C) $\frac { 1 } { 13 }$
D) 2
E) $\frac { 10 } { 7 }$

Question 50

Calculate the derivative of the function.
​

Question 51

Find the indicated derivative. $y = 19 x ^ { 3 } + \frac { 13 } { x }$ , $x = 14$ when $t = 1$ , $\left. \frac { \mathrm { d } x } { \mathrm {~d} t } \right| _ { z = 1 } = 20$ ; $\left. \frac { \mathrm { d } y } { \mathrm {~d} t } \right| _ { t = 1 } =$
Please round the answer to the nearest hundredth.

A) $\left. \frac { \mathrm { d } y } { \mathrm {~d} t } \right| _ { t = 1 } = 159586.73$
B) $\left. \frac { \mathrm { d } y } { \mathrm {~d} t } \right| _ { t = 1 } = 74461.43$
C) $\left. \frac { \mathrm { d } y } { \mathrm {~d} t } \right| _ { t = 1 } = 11171.93$
D) $\left. \frac { \mathrm { d } y } { \mathrm {~d} t } \right| _ { \imath = 1 } = 223438.67$
E) $\left. \frac { \mathrm { d } y } { \mathrm {~d} t } \right| _ { t = 1 } = 15958.67$

Question 52

Calculate the derivative of the function.
​ ​
Please enter your answer as an expression.

Question 53

Calculate $\frac { \mathrm { d } y } { \mathrm {~d} x }$ . You need not expand your answer. $y = ( \sqrt { x } + 2 ) \left( \sqrt { x } + \frac { 2 } { x ^ { 2 } } \right)$

A) $\frac { 1 } { 2 \sqrt { x } } \left( \sqrt { x } + \frac { 2 } { x ^ { 2 } } \right) + \left( \frac { 1 } { 2 \sqrt { x } } - \frac { 4 } { x ^ { 3 } } \right) ( \sqrt { x } + 2 )$
B) $\frac { \sqrt { x } } { 2 } \left( \sqrt { x } + \frac { 2 } { x ^ { 2 } } \right) + \left( \frac { \sqrt { x } } { 2 } - \frac { 4 } { x ^ { 3 } } \right) ( \sqrt { x } + 2 )$
C) $\frac { 1 } { 2 \sqrt { x } } \left( \sqrt { x } + \frac { 2 } { x ^ { 2 } } \right) + \left( \frac { 1 } { 2 \sqrt { x } } - 4 x \right) ( \sqrt { x } + 2 )$
D) $\frac { 1 } { \sqrt { x } } \left( \sqrt { x } + \frac { 2 } { x ^ { 2 } } \right) + \left( \frac { 1 } { \sqrt { x } } - \frac { 4 } { x } \right) ( \sqrt { x } + 2 )$
E) $\frac { 1 } { 2 \sqrt { x } } \left( \sqrt { x } + \frac { 2 } { x ^ { 2 } } \right) + \left( \frac { 1 } { 2 \sqrt { x } } + \frac { 4 } { x ^ { 3 } } \right) ( \sqrt { x } + 2 )$

Question 54

Calculate $\frac { \mathrm { d } y } { \mathrm {~d} x }$ . You need not expand your answer. $y = \left( \frac { x } { 1.2 } + \frac { 1.2 } { x } \right) \left( x ^ { 2 } + 7 \right)$

A) $2 x \left( \frac { 1 } { 1.2 } - \frac { 1.2 } { x ^ { 2 } } \right)$
B) $\left( \frac { 1 } { 1.2 } - \frac { 1.2 } { x ^ { 2 } } \right) \left( x ^ { 2 } + 7 \right) - 2 x \left( \frac { x } { 1.2 } + \frac { 1.2 } { x } \right)$
C) $2 x \left( \frac { 1 } { 1.2 } - \frac { 1.2 } { x ^ { 2 } } \right) + \left( \frac { x } { 1.2 } + \frac { 1.2 } { x } \right) \left( x ^ { 2 } + 7 \right)$
D) $2 x$
E) $\left( \frac { 1 } { 1.2 } - \frac { 1.2 } { x ^ { 2 } } \right) \left( x ^ { 2 } + 7 \right) + 2 x \left( \frac { x } { 1.2 } + \frac { 1.2 } { x } \right)$

Question 55

Find the indicated derivative. ? $y = 5 \sqrt { x } + \frac { 9 } { \sqrt { x } }$ , $x = 4$ when $t = 1$ , $\left. \frac { \mathrm { d } x } { \mathrm {~d} t } \right| _ { t = 1 } = 5$ ; $\left. \frac { \mathrm { d } y } { \mathrm {~d} t } \right| _ { t = 1 } =$
Please round the answer to the nearest hundredth.

A) $\left. \frac { \mathrm { d } y } { \mathrm {~d} t } \right| _ { t = 1 } = 3.44$
B) $\left. \frac { \mathrm { d } y } { \mathrm {~d} t } \right| _ { t = 1 } = 34.38$
C) $\left. \frac { \mathrm { d } y } { \mathrm {~d} t } \right| _ { t = 1 } = 11.88$
D) $\left. \frac { \mathrm { d } y } { \mathrm {~d} t } \right| _ { t = 1 } = 0.69$
E) $\left. \frac { \mathrm { d } y } { \mathrm {~d} t } \right| _ { t = 1 } = 6.88$

Question 56

Find the derivative of the function. $f ( x ) = 5 x$

A) $\chi$
B) 5
C) 4
D) $4 x$
E) $x ^ { 2 }$

Question 57

A mold culture in a dorm refrigerator is circular and growing. The radius is increasing at a rate of 0.1 cm/day. How fast is the area growing when the culture is 6 centimeters in radius (The area of a disc of radius r is $A = \pi r ^ { 2 }$ .)

A) $\frac { \mathrm { d } A } { \mathrm {~d} t } = 18 \pi \frac { \mathrm { cm } } { \mathrm { day } }$
B) $\frac { \mathrm { d } A } { \mathrm {~d} t } = 0.6 \pi \frac { \mathrm { cm } } { \mathrm { day } }$
C) $\frac { \mathrm { d } A } { \mathrm {~d} t } = 2.4 \pi \frac { \mathrm { cm } } { \mathrm { day } }$
D) $\frac { \mathrm { d } A } { \mathrm {~d} t } = 1.2 \pi \frac { \mathrm { cm } } { \mathrm { day } }$
E) $\frac { \mathrm { d } A } { \mathrm {~d} t } = 0.2 \pi \frac { \mathrm { cm } } { \mathrm { day } }$

Question 58

An offshore oil well is leaking oil and creating a circular oil slick. If the radius of the slick is growing at a rate of 7 miles per hour, find the rate at which the area is increasing when the radius is 3 miles. (The area of a disc of radius r is $A = \pi r ^ { 2 }$ .)

A) $\frac { \mathrm { d } A } { \mathrm {~d} t } = 9 \pi \frac { \mathrm { mi } ^ { 2 } } { \mathrm { hr } }$
B) $\frac { \mathrm { d } A } { \mathrm {~d} t } = 14 \pi \frac { \mathrm { mi } ^ { 2 } } { \mathrm { hr } }$
C) $\frac { \mathrm { d } A } { \mathrm {~d} t } = 42 \pi \frac { \mathrm { mi } ^ { 2 } } { \mathrm { hr } }$
D) $\frac { \mathrm { d } A } { \mathrm {~d} t } = 7 \pi \frac { \mathrm { mi } ^ { 2 } } { \mathrm { hr } }$
E) $\frac { \mathrm { d } A } { \mathrm {~d} t } = 21 \pi \frac { \mathrm { mi } ^ { 2 } } { \mathrm { hr } }$

Question 59

The soap bubble I am blowing has a radius that is growing at a rate of 3 cm/s. How fast is the surface area growing when the radius is 10 cm (The surface area of a sphere of radius r is $S = 4 \pi r ^ { 2 }$ .)

A) $\frac { \mathrm { d } S } { \mathrm {~d} t } = 85 \pi \frac { \mathrm { cm } ^ { 2 } } { \mathrm {~s} }$
B) $\frac { \mathrm { d } S } { \mathrm {~d} t } = \frac { 240 } { \pi } \frac { \mathrm { cm } ^ { 2 } } { \mathrm {~s} }$
C) $\frac { \mathrm { d } S } { \mathrm {~d} t } = 240 \pi \frac { \mathrm { cm } ^ { 2 } } { \mathrm {~s} }$
D) $\frac { \mathrm { d } S } { \mathrm {~d} t } = 243 \pi \frac { \mathrm { cm } ^ { 2 } } { \mathrm {~s} }$
E) $\frac { \mathrm { d } S } { \mathrm {~d} t } = 24 \pi \frac { \mathrm { cm } ^ { 2 } } { \mathrm {~s} }$

Question 60

Calculate $\frac { \mathrm { d } y } { \mathrm {~d} x }$ . You need not expand your answer. ? $y = \left( 9 x ^ { 2 } + x \right) \left( x - x ^ { 2 } \right)$

A) $( 18 x + 1 ) ( 1 - x ) + \left( x - x ^ { 2 } \right) \left( 9 x ^ { 2 } + x \right)$
B) $( 18 x + 1 ) \left( x - x ^ { 2 } \right) + ( 1 - 2 x ) \left( 9 x ^ { 2 } - x \right)$
C) $( 18 x + 1 ) \left( x - x ^ { 2 } \right) + ( 1 - 2 x ) \left( 9 x ^ { 2 } + x \right)$
D) $- 36 x ^ { 2 } + 20 x + 1$
E) $( 18 x + 1 ) ( 1 - x ) + \left( x - 2 x ^ { 2 } \right) \left( 9 x ^ { 2 } + x \right)$

Question 61

The monthly sales of Sunny Electronics' new stereo system is given by hundred units per month, x months after its introduction. The price Sunny charges is dollars per stereo system, x months after its introduction. The revenue Sunny earns then must be . Find the rate of change of revenue 10 months after introduction.
​
Please enter your answer in dollars/month without the units.

Question 62

The monthly sales of Sunny Electronics' new stereo system is given by $S ( x ) = 30 x - x ^ { 2 }$ hundred units per month, x months after its introduction. The price Sunny charges is $p ( x ) = 1,000 - x ^ { 2 }$ dollars per stereo system, x months after its introduction. The revenue Sunny earns then must be $R ( x ) = 100 p ( x ) S ( x )$ . Find the rate of change of revenue 6 months after introduction. Round your answer to the nearest dollar.

A)$437,700 per month
B) $43,770 per month
C) $1,562,400 per month
D) -$391,800 per month
E) -$322,700 per month

Question 63

The "Verhulst model" for population growth specifies the reproductive rate of an organism as a function of the total population according to the following formula: ? $R ( p ) = \frac { r } { 1 + k p }$
Where p is the total population in thousands of organisms, r and k are constants that depend on the particular circumstances and organism being studied, and R(p) is the reproduction rate in thousands of organisms per hour. If $k = 0.075$ and $r = 35$ , find $R ^ { \prime } ( p )$ .

A) $\frac { 2.625 } { 1 + 0.075 p ^ { 2 } }$
B) $\frac { 35 } { 1 + 0.075 p ^ { 2 } }$
C) $\frac { 2.625 } { ( 1 + 0.075 p ) ^ { 2 } }$
D) $\frac { 2.625 } { 1 + 0.075 p }$
E) $- \frac { 2.625 } { ( 1 + 0.075 p ) ^ { 2 } }$

Question 64

For the cost function C(x), find the marginal cost at the given production level x. Round your answer to two decimal places.
$C ( x ) = 25,000 + 20 x - \frac { x ^ { 2 } } { 1,000 } , x = 200$

A)$19.62 per item
B) $21.28 per item
C) $9.61 per item
D) $19.60 per item
E) $15.00 per item

Question 65

Calculate $\frac { \mathrm { d } y } { \mathrm {~d} x }$ . You need not expand your answer. $y = \frac { ( x + 2 ) ( x + 1 ) } { 2 x - 4 }$

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

Question 66

Your monthly profit (in dollars) from selling magazines is given by $P ( x ) = 5 x + 2 \sqrt { x }$ where x is the number of magazines you sell in a month. If you are currently selling $x = 50$ magazines per month, find your profit and your marginal profit.

A) $P ( 50 ) = \$ 132.07 , P ^ { \prime } ( 50 ) = \$ 2.57$
B) $P ( 50 ) = \$ 260.00 , P ^ { \prime } ( 50 ) = \$ 0.49$
C) $P ( 50 ) = \$ 528.28 , P ^ { \prime } ( 50 ) = \$ 5.64$
D) $P ( 50 ) = \$ 260.00 , P ^ { \prime } ( 50 ) = \$ 5.14$
E) $P ( 50 ) = \$ 264.14 , P ^ { \prime } ( 50 ) = \$ 5.14$

Question 67

Find the equation of the tangent line to the graph of the given function at the point with $x = 4$ . $f ( x ) = \frac { x + 4 } { x + 1 }$

A) $y = - 0.12 x$
B) $y = 0.12 x + 1.12$
C) $y = - 1.6$
D) $y = - 0.12 x + 2.08$
E) $y = 1.6$

Question 68

The Thoroughbred Bus Company finds that its monthly costs for one particular year were given by $C ( t ) = 100 + t ^ { 2 }$ dollars after t months. After t months, the company had $P ( t ) = 1,000 + t ^ { 2 }$ passengers per month. How fast was its cost per passenger changing after 6 months

Enter your answer in dollars/month rounded to the nearest cent and without the units.

Question 69

For the cost function, find the marginal cost at the given production level x. Round your answer to two decimal places.
$C ( x ) = 30,000 + 10 x - \frac { x ^ { 2 } } { 10,000 } , x = 2,000$

A)$9.60 per item
B) $9.58 per item
C) $10.00 per item
D) $9.61 per item
E) $10.40 per item

Question 70

Calculate $\frac { \mathrm { d } y } { \mathrm {~d} x }$ . You need not expand your answer. $y = \frac { 4.2 x ^ { - 0.8 } - 0.6 x ^ { - 0.5 } } { 0.4 + x ^ { 0.1 } }$

A) $\frac { \left( - 3.36 x ^ { - 1.8 } + 0.3 x ^ { - 1.5 } \right) \left( 0.4 + x ^ { 0.1 } \right) + 0.1 x ^ { - 0.9 } \left( 4.2 x ^ { 0.8 } - 0.6 x ^ { 0.5 } \right) } { \left( 0.4 + x ^ { 0.1 } \right) ^ { 2 } }$
B) $\frac { - 3.36 x ^ { - 1.8 } + 0.3 x ^ { - 1.5 } } { 0.4 + x ^ { 0.1 } }$
C) $\frac { \left( - 3.36 x ^ { - 1.8 } + 0.3 x ^ { - 1.5 } \right) \left( 0.4 + x ^ { 0.1 } \right) - 0.1 x ^ { - 0.9 } \left( 4.2 x ^ { - 0.8 } - 0.6 x ^ { - 0.5 } \right) } { \left( 0.1 x ^ { - 0.9 } \right) ^ { 2 } }$
D) $\frac { \left( - 3.36 x ^ { - 1.8 } + 0.3 x ^ { - 1.5 } \right) \left( 0.4 + x ^ { 0.1 } \right) - 0.1 x ^ { - 0.9 } \left( 4.2 x ^ { - 0.8 } - 0.6 x ^ { - 0.5 } \right) } { \left( 0.4 + x ^ { 0.1 } \right) ^ { 2 } }$
E) $\frac { \left( - 3.36 x ^ { - 1.8 } + 0.3 x ^ { - 1.5 } \right) \left( 0.4 + x ^ { 0.1 } \right) - 0.1 x ^ { - 0.9 } \left( 4.2 x ^ { 0.8 } - 0.6 x ^ { 0.5 } \right) } { 0.4 + x ^ { 0.1 } }$

Question 71

The Thoroughbred Bus Company finds that its monthly costs for one particular year were given by $C ( t ) = 100 + t ^ { 2 }$ dollars after t months. After t months, the company had $P ( t ) = 1,000 + t ^ { 2 }$ passengers per month. How fast was its cost per passenger changing after 4 months Round your answer to the nearest cent.

A)$0.29 per month
B) $0.01 per month
C) -$0.21 per month
D) $0.46 per month
E) $0.07 per month

Question 72

Find the equation of the line tangent to the graph of the given function at the point .
​

Question 73

Compute the derivative. ? $\frac { \mathrm { d } } { \mathrm { d } t } \left[ \left( t ^ { 2 } - t ^ { 0.5 } \right) \left( t ^ { 0.5 } + t ^ { - 0.5 } \right) \right] _ { t } = 1$

A)3
B) 0
C) 1
D) 7
E) -1

Question 74

Your Porche's gas mileage (in miles per gallon) is given as a function M(x) of speed x in miles per hour. $M ( x ) = \frac { 15 } { x + 3,735 x ^ { - 1 } }$
Calculate $M ^ { \prime } ( x )$ .

A) $\frac { 15 \left( 1 - \frac { 3,735 } { x ^ { 2 } } \right) } { \left( x + \frac { 3,735 } { x } \right) ^ { 2 } }$
B) $\frac { 15 } { 1 - \frac { 3,735 } { x ^ { 2 } } }$
C) $- \frac { 15 \left( x ^ { 2 } - 3,735 \right) } { \left( x ^ { 2 } + 3,735 \right) ^ { 2 } }$
D) $\frac { 15 } { \left( x + \frac { 3,735 } { x } \right) ^ { 2 } }$ ?
E) $\frac { 15 x } { x + 3,735 }$

Question 75

Find the equation of the line tangent to the graph of the given function at the point $x = 1$ . $f ( x ) = \left( x ^ { 3 } + 3 \right) \left( x ^ { 2 } + x \right)$

A) $y = 8$
B) $y = 18 x - 10$
C) $y = 8 x$
D) $y = 18$
E) $y = 18 x$

Question 76

Calculate $\frac { \mathrm { d } y } { \mathrm {~d} x }$ . You need not expand your answer. $y = \frac { 3 x - 6 } { ( x - 3 ) ( x - 1 ) ( x - 3 ) }$

A) $\frac { 3 ( x - 3 ) ( x - 1 ) ( x - 3 ) - \left( 3 x ^ { 2 } - 14 x + 15 \right) ( 3 x - 6 ) } { ( ( x - 3 ) ( x - 1 ) ( x - 3 ) ) ^ { 2 } }$
B) $\frac { 3 ( x - 3 ) ( x - 1 ) ( x - 3 ) - \left( 3 x ^ { 2 } - 14 x + 15 \right) ( 3 x - 6 ) } { ( x - 3 ) ( x - 1 ) ( x - 3 ) }$
C) $\frac { 3 ( x - 3 ) ( x - 1 ) ( x - 3 ) + \left( 3 x ^ { 2 } - 14 x + 7 \right) ( 3 x - 6 ) } { ( ( x - 3 ) ( x - 1 ) ( x - 3 ) ) ^ { 2 } }$
D) $\frac { 3 ( x - 3 ) ( x - 3 ) - \left( 3 x ^ { 2 } - 14 x + 15 \right) } { ( ( x - 3 ) ( x - 3 ) ) ^ { 2 } }$
E) $\frac { 3 } { 3 x ^ { 2 } - 14 x + 7 }$

Question 77

The cost, in thousands of dollars, of airing x television commercials during a Super Bowl game is given by the formula
$C ( x ) = 250 + 1,200 x - 0.005 x ^ { 2 }$ .

Estimate how fast (in dollars per television commercial) the cost is going up when $x = 8$ .

A)$1,200,080
B) $1,199,920
C) $1,199.92
D) $1,199,960
E) $1,200

Question 78

Compute the derivative. $\frac { \mathrm { d } } { \mathrm { d } x } \left[ \left( x ^ { 3 } + 2 x \right) \left( x ^ { 2 } - x \right) \right] _ { x = - 2 }$

A)-108
B) 132
C) 100
D) 131
E) 144

Question 79

The cost of producing x teddy bears per day at the Cuddly Companion Company is calculated by their marketing staff to be given by the formula
$C ( x ) = 200 + 60 x - 0.001 x ^ { 2 }$ .

Evaluate the average cost $\bar { C } ( 200 )$ .

A)$12,199.80
B) $59.80
C) $12,160.00
D) $59.20
E) $60.80

Question 80

Find the value of x for which the marginal profit is zero.
$C ( x ) = 2 x , R ( x ) = 6 x - \frac { x ^ { 2 } } { 1,000 }$

A) $x = 3,000$
B) $x = 4,000$
C) $x = 2,000$
D) $x = - 2,000$
E) $x = 1,000$