Question 1

Suppose that x and y are related by the equation $x ^ { 3 }$ + $( 2 y + 1 ) ^ { 2 }$ = $y ^ { 2 }$ . Use implicit differentiation to determine $\frac { d y } { d x }$ .

Accepted Answer

A)  $\frac { d y } { d x }$ = $\frac { 3 x ^ { 2 } + 4 ( 2 y + 1 ) } { 2 y }$ 
B)  $\frac { d y } { d x }$ = 3 $x ^ { 2 }$ + 2(y + 1) 
C)  $\frac { d y } { d x }$ = $\frac { - 3 x ^ { 2 } } { 2 ( 3 y + 2 ) }$ 
D)  $\frac { d y } { d x }$ = $\frac { - 3 x ^ { 2 } } { 6 y + 1 }$ 
A)  $\frac { d y } { d x }$ = $\frac { 3 x ^ { 2 } + 4 ( 2 y + 1 ) } { 2 y }$ 
B)  $\frac { d y } { d x }$ = 3 $x ^ { 2 }$ + 2(y + 1) 
C)  $\frac { d y } { d x }$ = $\frac { - 3 x ^ { 2 } } { 2 ( 3 y + 2 ) }$ 
D)  $\frac { d y } { d x }$ = $\frac { - 3 x ^ { 2 } } { 6 y + 1 }$

Question 2

Differentiate
-f(x) = $\frac { 6 x - 7 } { x ^ { 2 } + 5 }$ at x = -1
Enter just a reduced fraction.

Accepted Answer

The answer of Differentiate
-f(x) = \(\frac { 6 x -...

Question 3

The following function may be viewed as a composite function f(g(x)). Find f(x) and g(x). $\frac { 1 } { x ^ { 2 } + 2 x - 3 }$

Accepted Answer

A)  f(x) = x, G(x) = $x ^ { 2 }$ + 2x - 3 
B)  f(x) = $\frac { 1 } { x }$, G(x) = $\frac { 1 } { x ^ { 2 } + 2 x - 3 }$ 
C)  f(x) = $\frac { 1 } { x }$, G(x) = $x ^ { 2 }$ + 2x - 3 
D)  f(x) = $x ^ { 2 }$ + 2x - 3, G(x) = $\frac { 1 } { x }$ 
A)  f(x) = x, G(x) = $x ^ { 2 }$ + 2x - 3 
B)  f(x) = $\frac { 1 } { x }$, G(x) = $\frac { 1 } { x ^ { 2 } + 2 x - 3 }$ 
C)  f(x) = $\frac { 1 } { x }$, G(x) = $x ^ { 2 }$ + 2x - 3 
D)  f(x) = $x ^ { 2 }$ + 2x - 3, G(x) = $\frac { 1 } { x }$

Question 4

A publishing company has published a new magazine for young adults. The monthly sales S (in thousands) is given by $S ( t ) = \frac { 800 t } { t + 2 }$ where t is the number of months since the first issue was published. Find S(3) and S'(3) and interpret the results.

Accepted Answer

A)  At three months, the monthly sales are $480,000 and increasing at 64,000 magazines per month. 
B)  At three months, the monthly sales are $2, 400,000 and increasing at 64,000 magazines per month. 
C)  At three months, the monthly sales are $2,400,000 and increasing at 800,000 magazines per month. 
D)  At three months, the monthly sales are $480,000 and decreasing at 64,000 magazines per month. 
A)  At three months, the monthly sales are $480,000 and increasing at 64,000 magazines per month. 
B)  At three months, the monthly sales are $2, 400,000 and increasing at 64,000 magazines per month. 
C)  At three months, the monthly sales are $2,400,000 and increasing at 800,000 magazines per month. 
D)  At three months, the monthly sales are $480,000 and decreasing at 64,000 magazines per month.

Question 5

Find the slope of the tangent line to f(x) = $\frac { 1 } { x ^ { 2 } + 1 }$ at (-1, f(-1)).
Enter a reduced fraction.

Accepted Answer

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

Question 6

If y = 3u + 2 and u = $\frac { t } { t + 1 }$ , find $\frac { d y } { d t }$ .

Accepted Answer

A)  $\frac { 1 } { ( 3 u + 2 ) ^ { 2 } }$ 
B)  $\frac { 1 } { 9 }$ 
C)  $\frac { 1 } { ( t + 1 ) ^ { 2 } }$ 
D)  $\frac { 3 } { ( t + 1 ) ^ { 2 } }$ 
A)  $\frac { 1 } { ( 3 u + 2 ) ^ { 2 } }$ 
B)  $\frac { 1 } { 9 }$ 
C)  $\frac { 1 } { ( t + 1 ) ^ { 2 } }$ 
D)  $\frac { 3 } { ( t + 1 ) ^ { 2 } }$

Question 7

Use the chain rule to find the derivative of $\sqrt { 6 x ^ { 3 } - 2 x }$ at x = 1.
Enter just an integer.

Accepted Answer

The answer of Use the chain rule to find the...

Question 8

Differentiate
-f(x) = (4 $x ^ { 2 }$ + 4)(2 $x ^ { 2 }$ + 2x) at x = 1
Enter just an integer.

Accepted Answer

The answer of Differentiate
-f(x) = (4 \(x ^ { 2...

Question 9

Differentiate
-f(x) = ( $x ^ { 5 }$ - 8 $x ^ { 2 }$ + 5)( $x ^ { 3 }$ + $\sqrt { x }$ - 1) at x = 1
Enter just an integer.

Accepted Answer

The answer of Differentiate
-f(x) = ( \(x ^ { 5...

Question 10

Find all x-coordinates of points (x, y) on the graph of Q(x) = $\frac { x ^ { 3 } } { ( x + 2 ) ^ { 5 } }$ where the tangent line is horizontal. Enter your answer as just a, or a, b where these are integers and $\text { a }$ < b.

Accepted Answer

The answer of Find all x-coordinates of points (x, y)...

Question 11

Use implicit differentiation to determine $\frac { d y } { d x }$ where $x ^ { 3 }$ + $y ^ { 3 }$ = 2xy. Is $\frac { d y } { d x } = \frac { 2 y - 3 x ^ { 2 } } { 3 y ^ { 2 } - 2 x }$ correct?

Accepted Answer

A) True 
 B)False

Question 12

Differentiate
-f(x) = (2x + 1) $( 3 x - 2 ) ^ { 2 }$ at x = 1
Enter just an integer.

Accepted Answer

The answer of Differentiate
-f(x) = (2x + 1) \(( 3...

Question 13

Assume $\frac { 4 } { x }$ + $\sqrt { y }$ = x. What is the slope of the graph at the point (-1, 9)?

Accepted Answer

A)  30 
B)  4 + $\sqrt { 3 }$ 
C)  3 $\sqrt { 3 }$ 
D)  18 
A)  30 
B)  4 + $\sqrt { 3 }$ 
C)  3 $\sqrt { 3 }$ 
D)  18

Question 14

Find the slope of the tangent line to f(x) = $\frac { x ^ { 6 } + 4 x ^ { 3 } + 1 } { x ^ { 3 } + 1 }$ at (1, f(1)).
Enter just a reduced fraction of form $\frac { a } { b }$ .

Accepted Answer

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

Question 15

Let f(x) = $\frac { x ^ { 2 } - x } { x }$ , g(x) = $\frac { 1 } { \sqrt { x } }$ . Compute f(g(x)) at x= 4.
Enter just a reduced fraction.

Accepted Answer

The answer of Let f(x) = \(\frac { x ^...

Question 16

Differentiate
-f(x) = (-3 $x ^ { 7 }$ - $x ^ { 4 }$ )(2 $x ^ { 2 }$ - 5x + 3)

Accepted Answer

A)  -84 $x ^ { 7 }$ - 105 $x ^ { 6 }$ + 16 $x ^ { 4 }$ + 20 $x ^ { 3 }$ 
B)  -84 $x ^ { 7 }$ + 105 $x ^ { 6 }$ - 16 $x ^ { 4 }$ + 20 $x ^ { 3 }$ 
C)  -54 $x ^ { 8 }$ + 120 $x ^ { 7 }$ - 63 $x ^ { 6 }$ - 12 $x ^ { 5 }$ + 25 $x ^ { 4 }$ - 12 $x ^ { 3 }$ 
D)  30 $x ^ { 8 }$ - 90 $x ^ { 7 }$ + 63 $x ^ { 6 }$ + 4 $x ^ { 5 }$ - 15 $x ^ { 4 }$ + 12 $x ^ { 3 }$ 
A)  -84 $x ^ { 7 }$ - 105 $x ^ { 6 }$ + 16 $x ^ { 4 }$ + 20 $x ^ { 3 }$ 
B)  -84 $x ^ { 7 }$ + 105 $x ^ { 6 }$ - 16 $x ^ { 4 }$ + 20 $x ^ { 3 }$ 
C)  -54 $x ^ { 8 }$ + 120 $x ^ { 7 }$ - 63 $x ^ { 6 }$ - 12 $x ^ { 5 }$ + 25 $x ^ { 4 }$ - 12 $x ^ { 3 }$ 
D)  30 $x ^ { 8 }$ - 90 $x ^ { 7 }$ + 63 $x ^ { 6 }$ + 4 $x ^ { 5 }$ - 15 $x ^ { 4 }$ + 12 $x ^ { 3 }$

Question 17

If f(x) = $x ^ { 2 }$ - 9 and g(x) = $x ^ { 2 }$ - 16, find $\frac { \mathrm { d } } { \mathrm { dx } }$ g(f(x)).

Accepted Answer

A)  ( $x ^ { 2 }$ - 9 $) ^ { 2 }$ - 16 
B)  $4 x ^ { 3 }$ - 36x 
C)  (( $x ^ { 2 }$ - 9 $) ^ { 2 }$ - 16)(2x) 
D)  4 $x ^ { 3 }$ - 50 $x ^ { 2 }$ 
A)  ( $x ^ { 2 }$ - 9 $) ^ { 2 }$ - 16 
B)  $4 x ^ { 3 }$ - 36x 
C)  (( $x ^ { 2 }$ - 9 $) ^ { 2 }$ - 16)(2x) 
D)  4 $x ^ { 3 }$ - 50 $x ^ { 2 }$

Question 18

Differentiate
-f(x) = ( $x ^ { 3 }$ - 2 $) ^ { 3 }$ ( $x ^ { 3 }$ + 2 $) ^ { 4 }$ at x = -1
Enter just an integer.

Accepted Answer

The answer of Differentiate
-f(x) = ( \(x ^ { 3...

Question 19

Compute $\frac { d y } { d x }$ using the chain rule where y = $\frac { u ^ { 2 } } { u ^ { 2 } + 1 }$ and u = $\sqrt { 2 x + 4 }$ at x = 1.
Enter just a reduced fraction.

Accepted Answer

The answer of Compute \(\frac { d y } {...

Question 20

Differentiate
-f(x) = $\frac { x ^ { 2 } } { x + 1 }$ at x = 2
Enter just a reduced fraction.

Accepted Answer

The answer of Differentiate
-f(x) = \(\frac { x ^ {...

Suppose that x and y are related by the equation $x ^ { 3 }$ + $( 2 y + 1 ) ^ { 2 }$ = $y ^ { 2 }$ . Use implicit differentiation to determine $\frac { d y } { d x }$ .

Differentiate -f(x) = $\frac { 6 x - 7 } { x ^ { 2 } + 5 }$ at x = -1 Enter just a reduced fraction.

The following function may be viewed as a composite function f(g(x)). Find f(x) and g(x). $\frac { 1 } { x ^ { 2 } + 2 x - 3 }$

A publishing company has published a new magazine for young adults. The monthly sales S (in thousands) is given by $S ( t ) = \frac { 800 t } { t + 2 }$ where t is the number of months since the first issue was published. Find S(3) and S'(3) and interpret the results.

Find the slope of the tangent line to f(x) = $\frac { 1 } { x ^ { 2 } + 1 }$ at (-1, f(-1)). Enter a reduced fraction.

If y = 3u + 2 and u = $\frac { t } { t + 1 }$ , find $\frac { d y } { d t }$ .

Use the chain rule to find the derivative of $\sqrt { 6 x ^ { 3 } - 2 x }$ at x = 1. Enter just an integer.

Differentiate -f(x) = (4 $x ^ { 2 }$ + 4)(2 $x ^ { 2 }$ + 2x) at x = 1 Enter just an integer.

Differentiate -f(x) = ( $x ^ { 5 }$ - 8 $x ^ { 2 }$ + 5)( $x ^ { 3 }$ + $\sqrt { x }$ - 1) at x = 1 Enter just an integer.

Find all x-coordinates of points (x, y) on the graph of Q(x) = $\frac { x ^ { 3 } } { ( x + 2 ) ^ { 5 } }$ where the tangent line is horizontal. Enter your answer as just a, or a, b where these are integers and $\text { a }$ < b.

Use implicit differentiation to determine $\frac { d y } { d x }$ where $x ^ { 3 }$ + $y ^ { 3 }$ = 2xy. Is $\frac { d y } { d x } = \frac { 2 y - 3 x ^ { 2 } } { 3 y ^ { 2 } - 2 x }$ correct?

Differentiate -f(x) = (2x + 1) $( 3 x - 2 ) ^ { 2 }$ at x = 1 Enter just an integer.

Assume $\frac { 4 } { x }$ + $\sqrt { y }$ = x. What is the slope of the graph at the point (-1, 9)?

Find the slope of the tangent line to f(x) = $\frac { x ^ { 6 } + 4 x ^ { 3 } + 1 } { x ^ { 3 } + 1 }$ at (1, f(1)). Enter just a reduced fraction of form $\frac { a } { b }$ .

Let f(x) = $\frac { x ^ { 2 } - x } { x }$ , g(x) = $\frac { 1 } { \sqrt { x } }$ . Compute f(g(x)) at x= 4. Enter just a reduced fraction.

Differentiate -f(x) = (-3 $x ^ { 7 }$ - $x ^ { 4 }$ )(2 $x ^ { 2 }$ - 5x + 3)

If f(x) = $x ^ { 2 }$ - 9 and g(x) = $x ^ { 2 }$ - 16, find $\frac { \mathrm { d } } { \mathrm { dx } }$ g(f(x)).

Differentiate -f(x) = ( $x ^ { 3 }$ - 2 $) ^ { 3 }$ ( $x ^ { 3 }$ + 2 $) ^ { 4 }$ at x = -1 Enter just an integer.

Compute $\frac { d y } { d x }$ using the chain rule where y = $\frac { u ^ { 2 } } { u ^ { 2 } + 1 }$ and u = $\sqrt { 2 x + 4 }$ at x = 1. Enter just a reduced fraction.

Differentiate -f(x) = $\frac { x ^ { 2 } } { x + 1 }$ at x = 2 Enter just a reduced fraction.

The Derivative

Applications of the Derivative

Logarithm Functions

Applications of the Exponential and Natural Logarithm Functions

The Definite Integral

Functions of Several Variables

The Trigonometric Functions

Techniques of Integration

Differential Equations

Taylor Polynomials and Infinite Series

Probability and Calculus

Filters

Exam 3: Techniques of Differentiation

Suppose that x and y are related by the equation x3x ^ { 3 }x3 + (2y+1)2( 2 y + 1 ) ^ { 2 }(2y+1)2 = y2y ^ { 2 }y2 . Use implicit differentiation to determine dydx\frac { d y } { d x }dxdy​ .

Differentiate -f(x) = 6x−7x2+5\frac { 6 x - 7 } { x ^ { 2 } + 5 }x2+56x−7​ at x = -1 Enter just a reduced fraction.

The following function may be viewed as a composite function f(g(x)). Find f(x) and g(x). 1x2+2x−3\frac { 1 } { x ^ { 2 } + 2 x - 3 }x2+2x−31​

A publishing company has published a new magazine for young adults. The monthly sales S (in thousands) is given by S(t)=800tt+2S ( t ) = \frac { 800 t } { t + 2 }S(t)=t+2800t​ where t is the number of months since the first issue was published. Find S(3) and S'(3) and interpret the results.

Find the slope of the tangent line to f(x) = 1x2+1\frac { 1 } { x ^ { 2 } + 1 }x2+11​ at (-1, f(-1)). Enter a reduced fraction.

If y = 3u + 2 and u = tt+1\frac { t } { t + 1 }t+1t​ , find dydt\frac { d y } { d t }dtdy​ .

Use the chain rule to find the derivative of 6x3−2x\sqrt { 6 x ^ { 3 } - 2 x }6x3−2x​ at x = 1. Enter just an integer.

Differentiate -f(x) = (4 x2x ^ { 2 }x2 + 4)(2 x2x ^ { 2 }x2 + 2x) at x = 1 Enter just an integer.

Differentiate -f(x) = ( x5x ^ { 5 }x5 - 8 x2x ^ { 2 }x2 + 5)( x3x ^ { 3 }x3 + x\sqrt { x }x​ - 1) at x = 1 Enter just an integer.

Find all x-coordinates of points (x, y) on the graph of Q(x) = x3(x+2)5\frac { x ^ { 3 } } { ( x + 2 ) ^ { 5 } }(x+2)5x3​ where the tangent line is horizontal. Enter your answer as just a, or a, b where these are integers and a \text { a } a < b.

Use implicit differentiation to determine dydx\frac { d y } { d x }dxdy​ where x3x ^ { 3 }x3 + y3y ^ { 3 }y3 = 2xy. Is dydx=2y−3x23y2−2x\frac { d y } { d x } = \frac { 2 y - 3 x ^ { 2 } } { 3 y ^ { 2 } - 2 x }dxdy​=3y2−2x2y−3x2​ correct?

Differentiate -f(x) = (2x + 1) (3x−2)2( 3 x - 2 ) ^ { 2 }(3x−2)2 at x = 1 Enter just an integer.

Assume 4x\frac { 4 } { x }x4​ + y\sqrt { y }y​ = x. What is the slope of the graph at the point (-1, 9)?

Find the slope of the tangent line to f(x) = x6+4x3+1x3+1\frac { x ^ { 6 } + 4 x ^ { 3 } + 1 } { x ^ { 3 } + 1 }x3+1x6+4x3+1​ at (1, f(1)). Enter just a reduced fraction of form ab\frac { a } { b }ba​ .

Let f(x) = x2−xx\frac { x ^ { 2 } - x } { x }xx2−x​ , g(x) = 1x\frac { 1 } { \sqrt { x } }x​1​ . Compute f(g(x)) at x= 4. Enter just a reduced fraction.

Differentiate -f(x) = (-3 x7x ^ { 7 }x7 - x4x ^ { 4 }x4 )(2 x2x ^ { 2 }x2 - 5x + 3)

If f(x) = x2x ^ { 2 }x2 - 9 and g(x) = x2x ^ { 2 }x2 - 16, find ddx\frac { \mathrm { d } } { \mathrm { dx } }dxd​ g(f(x)).

Differentiate -f(x) = ( x3x ^ { 3 }x3 - 2 )3) ^ { 3 })3 ( x3x ^ { 3 }x3 + 2 )4) ^ { 4 })4 at x = -1 Enter just an integer.

Compute dydx\frac { d y } { d x }dxdy​ using the chain rule where y = u2u2+1\frac { u ^ { 2 } } { u ^ { 2 } + 1 }u2+1u2​ and u = 2x+4\sqrt { 2 x + 4 }2x+4​ at x = 1. Enter just a reduced fraction.

Differentiate -f(x) = x2x+1\frac { x ^ { 2 } } { x + 1 }x+1x2​ at x = 2 Enter just a reduced fraction.

The Derivative

Applications of the Derivative

Logarithm Functions

Applications of the Exponential and Natural Logarithm Functions

The Definite Integral

Functions of Several Variables

The Trigonometric Functions

Techniques of Integration

Differential Equations

Taylor Polynomials and Infinite Series

Probability and Calculus

Filters

Suppose that x and y are related by the equation $x ^ { 3 }$ + $( 2 y + 1 ) ^ { 2 }$ = $y ^ { 2 }$ . Use implicit differentiation to determine $\frac { d y } { d x }$ .

Differentiate -f(x) = $\frac { 6 x - 7 } { x ^ { 2 } + 5 }$ at x = -1 Enter just a reduced fraction.

The following function may be viewed as a composite function f(g(x)). Find f(x) and g(x). $\frac { 1 } { x ^ { 2 } + 2 x - 3 }$

A publishing company has published a new magazine for young adults. The monthly sales S (in thousands) is given by $S ( t ) = \frac { 800 t } { t + 2 }$ where t is the number of months since the first issue was published. Find S(3) and S'(3) and interpret the results.

Find the slope of the tangent line to f(x) = $\frac { 1 } { x ^ { 2 } + 1 }$ at (-1, f(-1)). Enter a reduced fraction.

If y = 3u + 2 and u = $\frac { t } { t + 1 }$ , find $\frac { d y } { d t }$ .

Use the chain rule to find the derivative of $\sqrt { 6 x ^ { 3 } - 2 x }$ at x = 1. Enter just an integer.

Differentiate -f(x) = (4 $x ^ { 2 }$ + 4)(2 $x ^ { 2 }$ + 2x) at x = 1 Enter just an integer.

Differentiate -f(x) = ( $x ^ { 5 }$ - 8 $x ^ { 2 }$ + 5)( $x ^ { 3 }$ + $\sqrt { x }$ - 1) at x = 1 Enter just an integer.

Find all x-coordinates of points (x, y) on the graph of Q(x) = $\frac { x ^ { 3 } } { ( x + 2 ) ^ { 5 } }$ where the tangent line is horizontal. Enter your answer as just a, or a, b where these are integers and $\text { a }$ < b.

Use implicit differentiation to determine $\frac { d y } { d x }$ where $x ^ { 3 }$ + $y ^ { 3 }$ = 2xy. Is $\frac { d y } { d x } = \frac { 2 y - 3 x ^ { 2 } } { 3 y ^ { 2 } - 2 x }$ correct?

Differentiate -f(x) = (2x + 1) $( 3 x - 2 ) ^ { 2 }$ at x = 1 Enter just an integer.

Assume $\frac { 4 } { x }$ + $\sqrt { y }$ = x. What is the slope of the graph at the point (-1, 9)?

Find the slope of the tangent line to f(x) = $\frac { x ^ { 6 } + 4 x ^ { 3 } + 1 } { x ^ { 3 } + 1 }$ at (1, f(1)). Enter just a reduced fraction of form $\frac { a } { b }$ .

Let f(x) = $\frac { x ^ { 2 } - x } { x }$ , g(x) = $\frac { 1 } { \sqrt { x } }$ . Compute f(g(x)) at x= 4. Enter just a reduced fraction.

Differentiate -f(x) = (-3 $x ^ { 7 }$ - $x ^ { 4 }$ )(2 $x ^ { 2 }$ - 5x + 3)

If f(x) = $x ^ { 2 }$ - 9 and g(x) = $x ^ { 2 }$ - 16, find $\frac { \mathrm { d } } { \mathrm { dx } }$ g(f(x)).

Differentiate -f(x) = ( $x ^ { 3 }$ - 2 $) ^ { 3 }$ ( $x ^ { 3 }$ + 2 $) ^ { 4 }$ at x = -1 Enter just an integer.

Compute $\frac { d y } { d x }$ using the chain rule where y = $\frac { u ^ { 2 } } { u ^ { 2 } + 1 }$ and u = $\sqrt { 2 x + 4 }$ at x = 1. Enter just a reduced fraction.

Differentiate -f(x) = $\frac { x ^ { 2 } } { x + 1 }$ at x = 2 Enter just a reduced fraction.