Question 1

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

Accepted Answer

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)    $6 e ^ { 4 x ^ { 6 } } x ^ { 5 } \ln ( 8 x ) + \frac { 8 e ^ { 4 x ^ { 6 } } } { x }$ 
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 2

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

Accepted Answer

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)  $\frac { 1 } { 2 \sqrt { x } } \left( \sqrt { x } + \frac { 2 } { x ^ { 2 } } \right) + \left( \frac { 1 } { 2 \sqrt { x } } + \frac { 4 } { x ^ { 3 } } \right) ( \sqrt { x } + 2 )$ 
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 3

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

Accepted Answer

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

Question 4

Use the shortcut rules to calculate the derivative of the given function.   $f ( x ) = - 3 x ^ { 0.75 }$

Accepted Answer

A)  $f ^ { \prime } ( x ) = - 3 x ^ { - 0.25 }$ 
B)  $f ^ { \prime } ( x ) = - 2.25 x$ 
C)  $f ^ { \prime } ( x ) = - 3 x ^ { 0.75 }$ 
D)  $f ^ { \prime } ( x ) = - 2.25 x ^ { 0.75 }$ 
E)  $f ^ { \prime } ( x ) = - 2.25 x ^ { - 0.25 }$ 
A)  $f ^ { \prime } ( x ) = - 3 x ^ { - 0.25 }$ 
B)  $f ^ { \prime } ( x ) = - 2.25 x$ 
C)  $f ^ { \prime } ( x ) = - 3 x ^ { 0.75 }$ 
D)  $f ^ { \prime } ( x ) = - 2.25 x ^ { 0.75 }$ 
E)  $f ^ { \prime } ( x ) = - 2.25 x ^ { - 0.25 }$

Question 5

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.

Accepted Answer

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)  $\frac { \mathrm { d } q } { \mathrm {~d} t } = 143 , \frac { \mathrm { d } R } { \mathrm {~d} q } = 900 , \frac { \mathrm { d } R } { \mathrm {~d} t } = 128,260$ 
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 6

Your company is planning to air a number of television commercials during the ABC television network's presentation of the Academy Awards. ABC is charging your company $725,000 per 30-second spot. Additional fixed costs (development and personnel costs) amount to $400,000, and the network has agreed to provide a discount of $D ( x ) = 25,000 \sqrt { x }$ for x television spots. Compute marginal cost $C ^ { \prime } ( 5 )$ and average cost $\bar { C } ( 5 )$ .

Accepted Answer

A)  $C ^ { \prime } ( 5 ) = \$ 718,410$ per spot; $\bar { C } ( 5 ) = \$ 793,820$ per spot 
B)  $C ^ { \prime } ( 5 ) = \$ 793,820$ per spot; $\bar { C } ( 5 ) = \$ 756,615$ per spot 
C)  $C ^ { \prime } ( 5 ) = \$ 719,410$ per spot; $\bar { C } ( 5 ) = \$ 793,720$ per spot 
D)  $C ^ { \prime } ( 5 ) = \$ 793,820$ per spot; $\bar { C } ( 5 ) = \$ 719,410$ per spot 
E)  $C ^ { \prime } ( 5 ) = \$ 719,410$ per spot; $\bar { C } ( 5 ) = \$ 793,820$ per spot 
A)  $C ^ { \prime } ( 5 ) = \$ 718,410$ per spot; $\bar { C } ( 5 ) = \$ 793,820$ per spot 
B)  $C ^ { \prime } ( 5 ) = \$ 793,820$ per spot; $\bar { C } ( 5 ) = \$ 756,615$ per spot 
C)  $C ^ { \prime } ( 5 ) = \$ 719,410$ per spot; $\bar { C } ( 5 ) = \$ 793,720$ per spot 
D)  $C ^ { \prime } ( 5 ) = \$ 793,820$ per spot; $\bar { C } ( 5 ) = \$ 719,410$ per spot 
E)  $C ^ { \prime } ( 5 ) = \$ 719,410$ per spot; $\bar { C } ( 5 ) = \$ 793,820$ per spot

Question 7

Your Porsche's gas mileage (in miles per gallon) is given as a function M(x) of speed x in miles per hour. It is found that 
 $M ^ { \prime } ( x ) = \frac { 5,625 x ^ { - 2 } - 1 } { \left( 5,625 ^ { - 1 } + x \right) ^ { 2 } }$ .
 
Find $M ^ { \prime } ( 55 )$ , $M ^ { \prime } ( 75 )$ and $M ^ { \prime } ( 95 )$ .

Accepted Answer

A)  $M ^ { \prime } ( 55 ) = 0.00004 , M ^ { \prime } ( 75 ) = - 0.00028 , M ^ { \prime } ( 95 ) = 0$ 
B)  $M ^ { \prime } ( 55 ) = 0.00028 , M ^ { \prime } ( 75 ) = 0 , M ^ { \prime } ( 95 ) = - 0.00004$ 
C)  $M ^ { \prime } ( 55 ) = 0.00004 , M ^ { \prime } ( 75 ) = 0 , M ^ { \prime } ( 95 ) = - 0.00028$ 
D)  $M ^ { \prime } ( 55 ) = 0 , M ^ { \prime } ( 75 ) = 0.00028 , M ^ { \prime } ( 95 ) = - 0.00004$ 
E)  $M ^ { \prime } ( 55 ) = 0 , M ^ { \prime } ( 75 ) = 0.00004 , M ^ { \prime } ( 95 ) = - 0.00004$ 
A)  $M ^ { \prime } ( 55 ) = 0.00004 , M ^ { \prime } ( 75 ) = - 0.00028 , M ^ { \prime } ( 95 ) = 0$ 
B)  $M ^ { \prime } ( 55 ) = 0.00028 , M ^ { \prime } ( 75 ) = 0 , M ^ { \prime } ( 95 ) = - 0.00004$ 
C)  $M ^ { \prime } ( 55 ) = 0.00004 , M ^ { \prime } ( 75 ) = 0 , M ^ { \prime } ( 95 ) = - 0.00028$ 
D)  $M ^ { \prime } ( 55 ) = 0 , M ^ { \prime } ( 75 ) = 0.00028 , M ^ { \prime } ( 95 ) = - 0.00004$ 
E)  $M ^ { \prime } ( 55 ) = 0 , M ^ { \prime } ( 75 ) = 0.00004 , M ^ { \prime } ( 95 ) = - 0.00004$

Question 8

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

Accepted Answer

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 
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 9

Assume that the demand function for tuna in a small coastal town is given by $p = \frac { 20,000 } { q ^ { 1.5 } }$ where p is the price (in dollars) per pound of tuna and q is the number of pounds of tuna that can be sold at the price p in 1 month. Calculate the price that the town's fishery should charge for tuna in order to produce a demand of 625 pounds of tuna per month.

Accepted Answer

A) $1 
B)  $2 
C)  $1,200 
D)  $800 
E)  $32 
A) $1 
B)  $2 
C)  $1,200 
D)  $800 
E)  $32

Question 10

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.

Accepted Answer

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

Question 11

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

Accepted Answer

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 
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 12

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

Accepted Answer

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)    $\frac { 2 } { ( 2 x + 8 ) } + \frac { 3 } { ( 3 x + 6 ) }$ 
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 13

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.

Accepted Answer

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 
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 14

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

Accepted Answer

A)  $y = - 0.12 x$ 
B)  $y = 0.12 x + 1.12$ 
C)  $y = - 1.6$ 
D)  $y = - 0.12 x + 2.08$ 
E)  $y = 1.6$ 
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 15

The oxygen consumption of a bird embryo increases from the time the egg is laid through the time the chick hatches. In a typical galliform bird, the oxygen consumption (in milliliters per hour) can be approximated by 
 $c ( t ) = 0.00279 t ^ { 3 } + 0.133 t ^ { 2 } - 0.893 t + 0.14 , ( 8 \leq t \leq 30 )$  
Where t is the time (in days) since the egg was laid. (An egg will typically hatch at around $t = 28$ .)
 
Find $c ^ { \prime } ( 26 )$ . Round your answer to two decimal places.

Accepted Answer

A)  $c ^ { \prime } ( 26 ) = 3.63$ 
B)  $c ^ { \prime } ( 26 ) = 8.22$ 
C)  $c ^ { \prime } ( 26 ) = 11.68$ 
D)  $c ^ { \prime } ( 26 ) = 0.36$ 
E)  $c ^ { \prime } ( 26 ) = - 2.15$ 
A)  $c ^ { \prime } ( 26 ) = 3.63$ 
B)  $c ^ { \prime } ( 26 ) = 8.22$ 
C)  $c ^ { \prime } ( 26 ) = 11.68$ 
D)  $c ^ { \prime } ( 26 ) = 0.36$ 
E)  $c ^ { \prime } ( 26 ) = - 2.15$

Question 16

Calculate the derivative of the function.
​ $f ( x ) = \left[ ( 8.5 x - 6 ) ^ { 6 } + ( 4.3 x + 6 ) ^ { 5 } \right] ^ { 7 }$ ​
Please enter your answer as an expression.

Accepted Answer

The answer of Calculate the derivative of the function.
​ \(f...

Question 17

Find the slope of the tangent line to the graph of the given function
 $f ( x ) = 7 x + 3$ at the indicated point $( 1,10 )$ .

Accepted Answer

A)  $f ^ { \prime } ( 1 ) = 1$ 
B)  $f ^ { \prime } ( 1 ) = 7$ 
C)  $f ^ { \prime } ( 1 ) = 10$ 
D)  $f ^ { \prime } ( 1 ) = 13$ 
E)  $f ^ { \prime } ( 1 ) = 4$ 
A)  $f ^ { \prime } ( 1 ) = 1$ 
B)  $f ^ { \prime } ( 1 ) = 7$ 
C)  $f ^ { \prime } ( 1 ) = 10$ 
D)  $f ^ { \prime } ( 1 ) = 13$ 
E)  $f ^ { \prime } ( 1 ) = 4$

Question 18

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

Accepted Answer

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 
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 19

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

Accepted Answer

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 
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 20

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

Accepted Answer

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

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

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

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

Use the shortcut rules to calculate the derivative of the given function. $f ( x ) = - 3 x ^ { 0.75 }$

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

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

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

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

Calculate the derivative of the function. $f ( x ) = \left[ ( 8.5 x - 6 ) ^ { 6 } + ( 4.3 x + 6 ) ^ { 5 } \right] ^ { 7 }$ Please enter your answer as an expression.

Find the slope of the tangent line to the graph of the given function $f ( x ) = 7 x + 3$ at the indicated point $( 1,10 )$ .

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

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

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

Functions and Applications

Nonlinear Functions and Models

The Mathematics of Finance

Systems of Linear Equations and Matrices

Matrix Algebra and Applications

Linear Programming

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Introduction to the Derivative

Applications of the Derivative

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Further Integration Techniques and Applications of the Integral

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Exam 11: Techniques of Differentiation

Find the derivative of the function. f(x)=e4x6ln⁡(8x)f ( x ) = e ^ { 4 x ^ { 6 } } \ln ( 8 x )f(x)=e4x6ln(8x)

Calculate dy dx\frac { \mathrm { d } y } { \mathrm {~d} x } dxdy​ . You need not expand your answer. y=(x+2)(x+2x2)y = ( \sqrt { x } + 2 ) \left( \sqrt { x } + \frac { 2 } { x ^ { 2 } } \right)y=(x​+2)(x​+x22​)

Find the value of x for which the marginal profit is zero. C(x)=2x,R(x)=6x−x21,000C ( x ) = 2 x , R ( x ) = 6 x - \frac { x ^ { 2 } } { 1,000 }C(x)=2x,R(x)=6x−1,000x2​

Use the shortcut rules to calculate the derivative of the given function. f(x)=−3x0.75f ( x ) = - 3 x ^ { 0.75 }f(x)=−3x0.75

Find the derivative of the function. r(x)=(e−6x6)8r ( x ) = \left( e ^ { - 6 x ^ { 6 } } \right) ^ { 8 }r(x)=(e−6x6)8

Find the derivative of the following function. f(x)=log⁡58xf ( x ) = \log _ { 5 } 8 xf(x)=log5​8x

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

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

Calculate the derivative of the function. ​ f(x)=[(8.5x−6)6+(4.3x+6)5]7f ( x ) = \left[ ( 8.5 x - 6 ) ^ { 6 } + ( 4.3 x + 6 ) ^ { 5 } \right] ^ { 7 }f(x)=[(8.5x−6)6+(4.3x+6)5]7 ​ Please enter your answer as an expression.

Find the slope of the tangent line to the graph of the given function f(x)=7x+3f ( x ) = 7 x + 3f(x)=7x+3 at the indicated point (1,10)( 1,10 )(1,10) .

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+10x−x210,000,x=2,000C ( x ) = 30,000 + 10 x - \frac { x ^ { 2 } } { 10,000 } , x = 2,000C(x)=30,000+10x−10,000x2​,x=2,000

Find the derivative of the function. r(x)=[ln⁡(x8)]3r ( x ) = \left[ \ln \left( x ^ { 8 } \right) \right] ^ { 3 }r(x)=[ln(x8)]3

Find dy dx\frac { \mathrm { d } y } { \mathrm {~d} x } dxdy​ using implicit differentiation. x2+y2=4x ^ { 2 } + y ^ { 2 } = 4x2+y2=4