Question 1

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

Accepted Answer

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 
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 2

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

Accepted Answer

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 
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 3

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 $x = 8$ .

Accepted Answer

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

Question 4

Given.
​ $\lim _ { x \rightarrow - 3 } \frac { x ^ { 2 } + 12 x + 27 } { x ^ { 2 } + 3 x }$ ​
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 -...

Question 5

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

Accepted Answer

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 
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 6

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

Accepted Answer

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)    $y = - \frac { 1 } { 10 } x - 2$ 
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 7

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

A) yes, 0 
B)  no, 0 
C)  no, 7 
D)  yes, 7 
E)  yes, 4 
A) yes, 0 
B)  no, 0 
C)  no, 7 
D)  yes, 7 
E)  yes, 4

Question 8

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

Accepted Answer

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

Question 9

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.

Accepted Answer

The answer of If $24,000 is invested in a savings...

Question 10

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

Accepted Answer

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 
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 11

Find the derivative of the following function.
​ $f ( x ) = \log _ { 4 } 8 x$

Accepted Answer

The answer of Find the derivative of the following function.
​...

Question 12

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

Accepted Answer

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

Question 13

Use logarithmic differentiation to find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ .
​ $y = x ^ { 5 x }$

Accepted Answer

The answer of Use logarithmic differentiation to find \(\frac {...

Question 14

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

Accepted Answer

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)  $\frac { \left( - 3.36 x ^ { - 1.8 } + 0.3 x ^ { - 1.5 } \right) \left( 0.4 + x ^ { 0.1 } \right) - 0.1 x ^ { - 0.9 } \left( 4.2 x ^ { 0.8 } - 0.6 x ^ { 0.5 } \right) } { 0.4 + x ^ { 0.1 } }$ 
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 15

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 $A = \pi r ^ { 2 }$ .)

Accepted Answer

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)  $\frac { \mathrm { d } A } { \mathrm {~d} t } = 21 \pi \frac { \mathrm { mi } ^ { 2 } } { \mathrm { hr } }$ 
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 16

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.

Accepted Answer

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 
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 17

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

Accepted Answer

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)   $\frac { \mathrm { d } y } { \mathrm {~d} x } = ( 5 x + 1 ) \left( \frac { 5 } { 5 x + 1 } + \frac { 7 } { 7 x - 1 } \right)$ 
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 18

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.

Accepted Answer

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)  $P ( 50 ) = \$ 264.14 , P ^ { \prime } ( 50 ) = \$ 5.14$ 
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 19

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 } =  $

Accepted Answer

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)   $\frac { \mathrm { d } y } { \mathrm {~d} t } = \left( 0.7 x ^ { 0.7 } + 2.7 x ^ { 0.7 } \right) \frac { \mathrm { d } x } { \mathrm {~d} t }$ 
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 20

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

Accepted Answer

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)    $\overline { x \ln x }$ 
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)    $\overline { x \ln x }$

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

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

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 $x = 8$ .

Given. $\lim _ { x \rightarrow - 3 } \frac { x ^ { 2 } + 12 x + 27 } { x ^ { 2 } + 3 x }$ 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 derivative of the function. $g ( x ) = \ln | 8 x - 9 |$

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

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

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.

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

Find the derivative of the following function. $f ( x ) = \log _ { 4 } 8 x$

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

Use logarithmic differentiation to find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ . $y = x ^ { 5 x }$

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

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 $A = \pi r ^ { 2 }$ .)

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.

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

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.

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

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

Functions and Applications

Nonlinear Functions and Models

The Mathematics of Finance

Systems of Linear Equations and Matrices

Matrix Algebra and Applications

Linear Programming

Sets and Counting

Probability

Random Variables and Statistics

Introduction to the Derivative

Applications of the Derivative

The Integral

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)=ln⁡∣(3x+5)2(7x+5)2(2x+8)f ( x ) = \ln \mid \frac { ( 3 x + 5 ) ^ { 2 } } { ( 7 x + 5 ) ^ { 2 } ( 2 x + 8 ) }f(x)=ln∣(7x+5)2(2x+8)(3x+5)2​

Find the derivative of the function. f(x)=(x3+5)ln⁡xf ( x ) = \left( x ^ { 3 } + 5 \right) \ln xf(x)=(x3+5)lnx

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

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

Given lim⁡x→7x2−14x+49x2−7x\lim _ { x \rightarrow 7 } \frac { x ^ { 2 } - 14 x + 49 } { x ^ { 2 } - 7 x }limx→7​x2−7xx2−14x+49​ 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 derivative of the function. f(x)=5xf ( x ) = 5 xf(x)=5x

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.

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+20x−x21,000,x=200C ( x ) = 25,000 + 20 x - \frac { x ^ { 2 } } { 1,000 } , x = 200C(x)=25,000+20x−1,000x2​,x=200

Find the derivative of the following function. ​ f(x)=log⁡48xf ( x ) = \log _ { 4 } 8 xf(x)=log4​8x

Find dy dx\frac { \mathrm { d } y } { \mathrm {~d} x } dxdy​ using implicit differentiation. 9x+10y=79 x + 10 y = 79x+10y=7 ?

Use logarithmic differentiation to find dy dx\frac { \mathrm { d } y } { \mathrm {~d} x } dxdy​ . ​ y=x5xy = x ^ { 5 x }y=x5x

Calculate dy dx\frac { \mathrm { d } y } { \mathrm {~d} x } dxdy​ . You need not expand your answer. y=4.2x−0.8−0.6x−0.50.4+x0.1y = \frac { 4.2 x ^ { - 0.8 } - 0.6 x ^ { - 0.5 } } { 0.4 + x ^ { 0.1 } }y=0.4+x0.14.2x−0.8−0.6x−0.5​

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=πr2A = \pi r ^ { 2 }A=πr2 .)

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.

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

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

Find the indicated derivative. The independent variable is a function of t. y=x0.7(1+x);dy dt=y = x ^ { 0.7 } ( 1 + x ) ; \frac { \mathrm { d } y } { \mathrm {~d} t } = y=x0.7(1+x); dtdy​=

Find dy dx\frac { \mathrm { d } y } { \mathrm {~d} x } dxdy​ using implicit differentiation. yln⁡x+y=2y \ln x + y = 2ylnx+y=2

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

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

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

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

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

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

Find the derivative of the following function. $f ( x ) = \log _ { 4 } 8 x$

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

Use logarithmic differentiation to find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ . $y = x ^ { 5 x }$

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

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 $A = \pi r ^ { 2 }$ .)

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

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.

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

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