Question 1

Find the slope of the tangent line to the graph of the given function $f ( x ) = 4 x ^ { 3 }$ at the point $( 2,32 )$ .

Accepted Answer

A)  $f ^ { \prime } ( 2 ) = 96$ 
B)  $f ^ { \prime } ( 2 ) = 0$ 
C)  $f ^ { \prime } ( 2 ) = 24$ 
D)  $f ^ { \prime } ( 2 ) = 48$ 
E)  $f ^ { \prime } ( 2 ) = 16$ 
A)  $f ^ { \prime } ( 2 ) = 96$ 
B)  $f ^ { \prime } ( 2 ) = 0$ 
C)  $f ^ { \prime } ( 2 ) = 24$ 
D)  $f ^ { \prime } ( 2 ) = 48$ 
E)  $f ^ { \prime } ( 2 ) = 16$

Question 2

Find the derivative of the function.
​ $k ( x ) = \frac { 6 x ^ { 12 } - 9 x ^ { 16 } } { x ^ { 6 } }$

Accepted Answer

The answer of Find the derivative of the function.
​ \(k...

Question 3

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

Accepted Answer

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)    $\frac { s e ^ { s t } } { 9 s - t e ^ { s t } }$ 
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 4

Given.
​ $\lim _ { x \rightarrow 7 } \frac { x ^ { 2 } - 14 x + 49 } { x ^ { 2 } - 7 x }$ ​
Say whether L'Hospital's rule applies.
​
__________
​
If it does, use it to evaluate the given limit. If not, use some other method.
​
__________

Accepted Answer

The answer of Given.
​ \(\lim _ { x \rightarrow 7...

Question 5

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

Accepted Answer

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

Question 6

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

Accepted Answer

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)    $s ^ { \prime } ( x ) = 6 \left( \frac { 8 x + 5 } { 2 x - 6 } \right) ^ { 5 }$ 
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 7

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

Accepted Answer

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 
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 8

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 $\bar { C } ( 200 )$ .

Accepted Answer

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

Question 9

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 $M ^ { \prime } ( x )$ .

Accepted Answer

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)  $\frac { 15 x } { x + 3,735 }$ 
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 10

Find the derivative of the function. 
$r ( x ) = \frac { 3 x } { 6 } - \frac { x ^ { 0.6 } } { 2 } + \frac { 8 } { 6 x ^ { 1.6 } } - 8$

Accepted Answer

A)  $r ^ { \prime } ( x ) = \frac { 3 } { 6 } - \frac { 0.6 } { 2 x ^ { 0.4 } } + \frac { 12.8 } { 6 x ^ { 2.6 } }$ 
B)  $r ^ { \prime } ( x ) = \frac { 3 } { 6 } - \frac { 0.6 x ^ { 0.4 } } { 2 } + \frac { 12.8 x ^ { 2.6 } } { 6 }$ 
C)  $r ^ { \prime } ( x ) = \frac { 3 } { 6 } - \frac { 0.6 } { 2 x ^ { 0.4 } } - \frac { 12.8 } { 6 x ^ { 2.6 } }$ 
D)  $r ^ { \prime } ( x ) = \frac { 3 } { 6 } - \frac { 0.6 } { 2 x ^ { 0.6 } } - \frac { 12.8 } { 6 x ^ { 1.6 } }$ 
E)  $r ^ { \prime } ( x ) = \frac { 3 } { 6 } - \frac { 0.6 x ^ { 1.6 } } { 2 } - \frac { 12.8 } { 6 x ^ { 0.6 } }$ 
A)  $r ^ { \prime } ( x ) = \frac { 3 } { 6 } - \frac { 0.6 } { 2 x ^ { 0.4 } } + \frac { 12.8 } { 6 x ^ { 2.6 } }$ 
B)  $r ^ { \prime } ( x ) = \frac { 3 } { 6 } - \frac { 0.6 x ^ { 0.4 } } { 2 } + \frac { 12.8 x ^ { 2.6 } } { 6 }$ 
C)  $r ^ { \prime } ( x ) = \frac { 3 } { 6 } - \frac { 0.6 } { 2 x ^ { 0.4 } } - \frac { 12.8 } { 6 x ^ { 2.6 } }$ 
D)  $r ^ { \prime } ( x ) = \frac { 3 } { 6 } - \frac { 0.6 } { 2 x ^ { 0.6 } } - \frac { 12.8 } { 6 x ^ { 1.6 } }$ 
E)  $r ^ { \prime } ( x ) = \frac { 3 } { 6 } - \frac { 0.6 x ^ { 1.6 } } { 2 } - \frac { 12.8 } { 6 x ^ { 0.6 } }$

Question 11

Given.  $\lim _ { x \rightarrow - 2 } \frac { x ^ { 2 } + 8 x + 12 } { x ^ { 2 } + 2 x }$  
Say whether L'Hospital's rule applies. If it does, use it to evaluate the given limit. If not, use some other method.

Accepted Answer

A) yes, 6 
B)  no, 2 
C)  yes, -2 
D)  no, -2 
E)  no, -6 
A) yes, 6 
B)  no, 2 
C)  yes, -2 
D)  no, -2 
E)  no, -6

Question 12

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

Accepted Answer

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)  $\frac { 3 } { 3 x ^ { 2 } - 14 x + 7 }$ 
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 13

Evaluate the following derivative.   $\frac { \mathrm { d } } { \mathrm { d } t } \left( p t ^ { 2 } + j t + r \right)$

Accepted Answer

A)  $2 p t + r$ 
B)  $2 p + j$ 
C)  $2 p t$ 
D)  $2 p t + j$ 
E)  $p t + j$ 
A)  $2 p t + r$ 
B)  $2 p + j$ 
C)  $2 p t$ 
D)  $2 p t + j$ 
E)  $p t + j$

Question 14

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

Accepted Answer

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 
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 15

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

Accepted Answer

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 
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 16

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

Accepted Answer

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

Question 17

Find all the values of x (if any) where the tangent line to the graph of the given equation is horizontal. Round your answer to two decimal places. 
$y = 10 x ^ { 2 } + 11 x + 5$

Accepted Answer

A)  $x = - 0.55$ 
B)  $x = - 5.5$ 
C)  $x = 0.55$ 
D)  $x = 5.5$ 
E)  $x = 0$ 
A)  $x = - 0.55$ 
B)  $x = - 5.5$ 
C)  $x = 0.55$ 
D)  $x = 5.5$ 
E)  $x = 0$

Question 18

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 $R ^ { \prime } ( p )$ .

Accepted Answer

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)  $- \frac { 2.625 } { ( 1 + 0.075 p ) ^ { 2 } }$ 
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 19

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.

Accepted Answer

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

Question 20

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

Accepted Answer

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

Find the slope of the tangent line to the graph of the given function $f ( x ) = 4 x ^ { 3 }$ at the point $( 2,32 )$ .

Find the derivative of the function. $k ( x ) = \frac { 6 x ^ { 12 } - 9 x ^ { 16 } } { x ^ { 6 } }$

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

Given. $\lim _ { x \rightarrow 7 } \frac { x ^ { 2 } - 14 x + 49 } { x ^ { 2 } - 7 x }$ Say whether L'Hospital's rule applies. If it does, use it to evaluate the given limit. If not, use some other method.

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

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

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

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 $\bar { C } ( 200 )$ .

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 $M ^ { \prime } ( x )$ .

Find the derivative of the function. $r ( x ) = \frac { 3 x } { 6 } - \frac { x ^ { 0.6 } } { 2 } + \frac { 8 } { 6 x ^ { 1.6 } } - 8$

Given. $\lim _ { x \rightarrow - 2 } \frac { x ^ { 2 } + 8 x + 12 } { x ^ { 2 } + 2 x }$ Say whether L'Hospital's rule applies. If it does, use it to evaluate the given limit. If not, use some other method.

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

Evaluate the following derivative. $\frac { \mathrm { d } } { \mathrm { d } t } \left( p t ^ { 2 } + j t + r \right)$

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

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

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

Find all the values of x (if any) where the tangent line to the graph of the given equation is horizontal. Round your answer to two decimal places. $y = 10 x ^ { 2 } + 11 x + 5$

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

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Exam 11: Techniques of Differentiation

Find the slope of the tangent line to the graph of the given function f(x)=4x3f ( x ) = 4 x ^ { 3 }f(x)=4x3 at the point (2,32)( 2,32 )(2,32) .

Find the derivative of the function. ​ k(x)=6x12−9x16x6k ( x ) = \frac { 6 x ^ { 12 } - 9 x ^ { 16 } } { x ^ { 6 } }k(x)=x66x12−9x16​

Find ds dt\frac { \mathrm { d } s } { \mathrm {~d} t } dtds​ using implicit differentiation. est=s9e ^ { s t } = s ^ { 9 }est=s9

Find the equation of the line tangent to the graph of the given function at the point x=1x = 1x=1 . f(x)=(x3+3)(x2+x)f ( x ) = \left( x ^ { 3 } + 3 \right) \left( x ^ { 2 } + x \right)f(x)=(x3+3)(x2+x)

Calculate the derivative of the function. s(x)=(8x+52x−6)6s ( x ) = \left( \frac { 8 x + 5 } { 2 x - 6 } \right) ^ { 6 }s(x)=(2x−68x+5​)6

Find the derivative of the function. e−5x5xe5x\frac { e ^ { - 5 x } } { 5 x e ^ { 5 x } }5xe5xe−5x​

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+60x−0.001x2C ( x ) = 200 + 60 x - 0.001 x ^ { 2 }C(x)=200+60x−0.001x2 . Evaluate the average cost Cˉ(200)\bar { C } ( 200 )Cˉ(200) .

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)=15x+3,735x−1M ( x ) = \frac { 15 } { x + 3,735 x ^ { - 1 } }M(x)=x+3,735x−115​ Calculate M′(x)M ^ { \prime } ( x )M′(x) .

Find the derivative of the function. r(x)=3x6−x0.62+86x1.6−8r ( x ) = \frac { 3 x } { 6 } - \frac { x ^ { 0.6 } } { 2 } + \frac { 8 } { 6 x ^ { 1.6 } } - 8r(x)=63x​−2x0.6​+6x1.68​−8

Given. lim⁡x→−2x2+8x+12x2+2x\lim _ { x \rightarrow - 2 } \frac { x ^ { 2 } + 8 x + 12 } { x ^ { 2 } + 2 x }limx→−2​x2+2xx2+8x+12​ Say whether L'Hospital's rule applies. If it does, use it to evaluate the given limit. If not, use some other method.

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

Evaluate the following derivative. ddt(pt2+jt+r)\frac { \mathrm { d } } { \mathrm { d } t } \left( p t ^ { 2 } + j t + r \right)dtd​(pt2+jt+r)

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

Find the derivative of the function. h(x)=e9x2−4x+1xh ( x ) = e ^ { 9 x ^ { 2 } - 4 x + \frac { 1 } { x } }h(x)=e9x2−4x+x1​

Find dy dx\frac { \mathrm { d } y } { \mathrm {~d} x } dxdy​ using implicit differentiation. x2−6y=7x ^ { 2 } - 6 y = 7x2−6y=7 ?

Find all the values of x (if any) where the tangent line to the graph of the given equation is horizontal. Round your answer to two decimal places. y=10x2+11x+5y = 10 x ^ { 2 } + 11 x + 5y=10x2+11x+5

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

Functions and Applications

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Find the slope of the tangent line to the graph of the given function $f ( x ) = 4 x ^ { 3 }$ at the point $( 2,32 )$ .

Find the derivative of the function. $k ( x ) = \frac { 6 x ^ { 12 } - 9 x ^ { 16 } } { x ^ { 6 } }$

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

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

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

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

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 $\bar { C } ( 200 )$ .

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 $M ^ { \prime } ( x )$ .

Find the derivative of the function. $r ( x ) = \frac { 3 x } { 6 } - \frac { x ^ { 0.6 } } { 2 } + \frac { 8 } { 6 x ^ { 1.6 } } - 8$

Given. $\lim _ { x \rightarrow - 2 } \frac { x ^ { 2 } + 8 x + 12 } { x ^ { 2 } + 2 x }$ Say whether L'Hospital's rule applies. If it does, use it to evaluate the given limit. If not, use some other method.

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

Evaluate the following derivative. $\frac { \mathrm { d } } { \mathrm { d } t } \left( p t ^ { 2 } + j t + r \right)$

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

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

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

Find all the values of x (if any) where the tangent line to the graph of the given equation is horizontal. Round your answer to two decimal places. $y = 10 x ^ { 2 } + 11 x + 5$

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