Question 1

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

Accepted Answer

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)  $3 x ^ { 2 } + 48 x + 8$ 
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)  $3 x ^ { 2 } + 48 x + 8$ B

Question 2

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

Accepted Answer

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

Question 3

Find the derivative of the function. 
 $g ( x ) = 6 x ^ { - 3 } + 2 x ^ { - 5 }$

Accepted Answer

A)  $g ^ { \prime } ( x ) = 18 x ^ { - 4 } + 10 x ^ { - 6 }$ 
B)  $g ^ { \prime } ( x ) = 6 x ^ { - 3 } + 2 x ^ { - 5 }$ 
C)  $g ^ { \prime } ( x ) = - 18 x ^ { - 3 } - 10 x ^ { - 5 }$ 
D)  $g ^ { \prime } ( x ) = - 6 x ^ { - 4 } - 2 x ^ { - 6 }$ 
E)  $g ^ { \prime } ( x ) = - 18 x ^ { - 4 } - 10 x ^ { - 6 }$ 
A)  $g ^ { \prime } ( x ) = 18 x ^ { - 4 } + 10 x ^ { - 6 }$ 
B)  $g ^ { \prime } ( x ) = 6 x ^ { - 3 } + 2 x ^ { - 5 }$ 
C)  $g ^ { \prime } ( x ) = - 18 x ^ { - 3 } - 10 x ^ { - 5 }$ 
D)  $g ^ { \prime } ( x ) = - 6 x ^ { - 4 } - 2 x ^ { - 6 }$ 
E)  $g ^ { \prime } ( x ) = - 18 x ^ { - 4 } - 10 x ^ { - 6 }$ E

Question 4

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

Accepted Answer

A)  $\frac { x y } { x ^ { 2 } + 1 }$ 
B)    $- \frac { x } { y }$ 
C)    $- \frac { x y } { x ^ { 2 } + 1 }$ 
D)    $2 y + 2 x$ 
E)    $- \frac { x ^ { 2 } + 1 } { x y }$ 
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 5

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.

Accepted Answer

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 
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 6

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.

Accepted Answer

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)  $\left. \frac { \mathrm { d } y } { \mathrm {~d} t } \right| _ { t = 1 } = 15958.67$ 
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 7

Find the slope of the tangent line to the graph of the given function at the indicated point.   $g ( t ) = \frac { 8 } { t ^ { 3 } } , ( 1.5,57 )$

Accepted Answer

A)  $g ^ { \prime } ( 1.5 ) = - 108$ 
B)  $g ^ { \prime } ( 1.5 ) = - 36$ 
C)  $g ^ { \prime } ( 1.5 ) = 54$ 
D)  $g ^ { \prime } ( 1.5 ) = - 27$ 
E)  $g ^ { \prime } ( 1.5 ) = - 54$ 
A)  $g ^ { \prime } ( 1.5 ) = - 108$ 
B)  $g ^ { \prime } ( 1.5 ) = - 36$ 
C)  $g ^ { \prime } ( 1.5 ) = 54$ 
D)  $g ^ { \prime } ( 1.5 ) = - 27$ 
E)  $g ^ { \prime } ( 1.5 ) = - 54$

Question 8

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

Accepted Answer

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

Question 9

According to the equation given, a toy manufacturer calculates its daily profit P (in dollars) based on the number of workers n it employs. 
$P = 300 n - 0.6 n ^ { 2 }$ 
Calculate the marginal product at an employment level of 100 workers.

Accepted Answer

A) $360 
B)  $24,000 
C)  $29,940 
D)  $180 
E)  $2,999,940 
A) $360 
B)  $24,000 
C)  $29,940 
D)  $180 
E)  $2,999,940

Question 10

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

Accepted Answer

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

Question 11

Find the derivative of the function.
​ $s ( x ) = 4 \sqrt { x } + \frac { 30 } { \sqrt { x } }$

Accepted Answer

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

Question 12

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

Accepted Answer

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)  $( 18 x + 1 ) ( 1 - x ) + \left( x - 2 x ^ { 2 } \right) \left( 9 x ^ { 2 } + x \right)$ 
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 13

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.

Accepted Answer

The answer of The Thoroughbred Bus Company finds that its...

Question 14

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

Accepted Answer

A)  $\frac { 10 - y } { x - 11 }$ 
B)    $\frac { 10 - x } { y - 11 }$ 
C)   $11 - y$ 
D)    $\frac { x - 11 } { 10 - y }$ 
E)    $x - 10$ 
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 15

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

Accepted Answer

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)    $\frac { x e ^ { y } - e ^ { x } } { y e ^ { x } - e ^ { y } }$ 
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 16

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.

Accepted Answer

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 
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 17

Calculate the derivative of the function.
​ $f ( x ) = \left( x ^ { 2 } + 5 x \right) ^ { 3 }$

Accepted Answer

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

Question 18

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

Accepted Answer

The answer of Find the equation of the line tangent...

Question 19

The estimated marginal revenue for sales of ESU soccer team T-shirts is given by 
 $R ^ { \prime } ( p ) = \frac { ( 6 - 2 p ) e ^ { - p ^ { 2 } + 7 p } } { 5,000,000 }$  
Where p is the price (in dollars) the soccer player charge for each shirt. Find $R ^ { \prime } ( 2 ) , R ^ { \prime } ( 3 ) , R ^ { \prime } ( 5 )$ .
 
Select the correct answer rounded to four decimal places.

Accepted Answer

A)  $R ^ { \prime } ( 2 ) = 0.0088 , R ^ { \prime } ( 3 ) = 0 , R ^ { \prime } ( 5 ) = - 0.0176$ 
B)  $R ^ { \prime } ( 3 ) = 0.0088 , R ^ { \prime } ( 5 ) = 0 , R ^ { \prime } ( 2 ) = - 0.0176$ 
C)  $R ^ { \prime } ( 2 ) = 0.0176 , R ^ { \prime } ( 3 ) = - 0.0088 , R ^ { \prime } ( 5 ) = 0$ 
D)  $R ^ { \prime } ( 2 ) = - 0.0176 , R ^ { \prime } ( 3 ) = 0 , R ^ { \prime } ( 5 ) = - 0.0088$ 
E)  $R ^ { \prime } ( 3 ) = 0.0088 , R ^ { \prime } ( 2 ) = 0 , R ^ { \prime } ( 5 ) = - 0.0176$ 
A)  $R ^ { \prime } ( 2 ) = 0.0088 , R ^ { \prime } ( 3 ) = 0 , R ^ { \prime } ( 5 ) = - 0.0176$ 
B)  $R ^ { \prime } ( 3 ) = 0.0088 , R ^ { \prime } ( 5 ) = 0 , R ^ { \prime } ( 2 ) = - 0.0176$ 
C)  $R ^ { \prime } ( 2 ) = 0.0176 , R ^ { \prime } ( 3 ) = - 0.0088 , R ^ { \prime } ( 5 ) = 0$ 
D)  $R ^ { \prime } ( 2 ) = - 0.0176 , R ^ { \prime } ( 3 ) = 0 , R ^ { \prime } ( 5 ) = - 0.0088$ 
E)  $R ^ { \prime } ( 3 ) = 0.0088 , R ^ { \prime } ( 2 ) = 0 , R ^ { \prime } ( 5 ) = - 0.0176$

Question 20

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

Accepted Answer

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)  $\frac { \mathrm { d } A } { \mathrm {~d} t } = 0.2 \pi \frac { \mathrm { cm } } { \mathrm { day } }$ 
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)  $\frac { \mathrm { d } A } { \mathrm {~d} t } = 0.2 \pi \frac { \mathrm { cm } } { \mathrm { day } }$

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

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

Find the derivative of the function. $g ( x ) = 6 x ^ { - 3 } + 2 x ^ { - 5 }$

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

Find the slope of the tangent line to the graph of the given function at the indicated point. $g ( t ) = \frac { 8 } { t ^ { 3 } } , ( 1.5,57 )$

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

According to the equation given, a toy manufacturer calculates its daily profit P (in dollars) based on the number of workers n it employs. $P = 300 n - 0.6 n ^ { 2 }$ Calculate the marginal product at an employment level of 100 workers.

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

Find the derivative of the function. $s ( x ) = 4 \sqrt { x } + \frac { 30 } { \sqrt { x } }$

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

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

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

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.

Calculate the derivative of the function. $f ( x ) = \left( x ^ { 2 } + 5 x \right) ^ { 3 }$

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

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

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Exam 11: Techniques of Differentiation

Calculate dy dx\frac { \mathrm { d } y } { \mathrm {~d} x } dxdy​ . y=x2(3x+2)(5x+2)y = x ^ { 2 } ( 3 x + 2 ) ( 5 x + 2 )y=x2(3x+2)(5x+2)

Find dy dx\frac { \mathrm { d } y } { \mathrm {~d} x } dxdy​ using implicit differentiation. exy=4e ^ { x } y = 4exy=4 ?

Find the derivative of the function. g(x)=6x−3+2x−5g ( x ) = 6 x ^ { - 3 } + 2 x ^ { - 5 }g(x)=6x−3+2x−5

Find dx dy\frac { \mathrm { d } x } { \mathrm {~d} y } dydx​ using implicit differentiation. (xy)2+y2=5( x y ) ^ { 2 } + y ^ { 2 } = 5(xy)2+y2=5

Find the slope of the tangent line to the graph of the given function at the indicated point. g(t)=8t3,(1.5,57)g ( t ) = \frac { 8 } { t ^ { 3 } } , ( 1.5,57 )g(t)=t38​,(1.5,57)

Compute the indicated derivative using the chain rule. y=10x2−7x;dx dy∣x=2y = 10 x ^ { 2 } - 7 x ; \left. \frac { \mathrm { d } x } { \mathrm {~d} y } \right| _ { x = 2 }y=10x2−7x; dydx​​x=2​

According to the equation given, a toy manufacturer calculates its daily profit P (in dollars) based on the number of workers n it employs. P=300n−0.6n2P = 300 n - 0.6 n ^ { 2 }P=300n−0.6n2 Calculate the marginal product at an employment level of 100 workers.

Find the derivative of the function. f(x)=4x3f ( x ) = \frac { 4 } { x ^ { 3 } }f(x)=x34​

Find the derivative of the function. ​ s(x)=4x+30xs ( x ) = 4 \sqrt { x } + \frac { 30 } { \sqrt { x } }s(x)=4x​+x​30​

Calculate dy dx\frac { \mathrm { d } y } { \mathrm {~d} x } dxdy​ . You need not expand your answer. ? y=(9x2+x)(x−x2)y = \left( 9 x ^ { 2 } + x \right) \left( x - x ^ { 2 } \right)y=(9x2+x)(x−x2)

Find dy dx\frac { \mathrm { d } y } { \mathrm {~d} x } dxdy​ using implicit differentiation. 10x+11y=xy10 x + 11 y = x y10x+11y=xy ?

Find dy dx\frac { \mathrm { d } y } { \mathrm {~d} x } dxdy​ using implicit differentiation. xey−yex=12x e ^ { y } - y e ^ { x } = 12xey−yex=12

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.

Calculate the derivative of the function. ​ f(x)=(x2+5x)3f ( x ) = \left( x ^ { 2 } + 5 x \right) ^ { 3 }f(x)=(x2+5x)3

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

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=πr2A = \pi r ^ { 2 }A=πr2 .)

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

Functions of Several Variables

Trigonometric Models

Filters

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

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

Find the derivative of the function. $g ( x ) = 6 x ^ { - 3 } + 2 x ^ { - 5 }$

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

Find the slope of the tangent line to the graph of the given function at the indicated point. $g ( t ) = \frac { 8 } { t ^ { 3 } } , ( 1.5,57 )$

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

According to the equation given, a toy manufacturer calculates its daily profit P (in dollars) based on the number of workers n it employs. $P = 300 n - 0.6 n ^ { 2 }$ Calculate the marginal product at an employment level of 100 workers.

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

Find the derivative of the function. $s ( x ) = 4 \sqrt { x } + \frac { 30 } { \sqrt { x } }$

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

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

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

Calculate the derivative of the function. $f ( x ) = \left( x ^ { 2 } + 5 x \right) ^ { 3 }$

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

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