Question 1

Use implicit differentiation to determine the slope of the graph of $\sqrt { x + y }$ = x at (2, 2).
Enter just an integer.

Accepted Answer

The answer of Use implicit differentiation to determine the slope...

Question 2

Use implicit differentiation to determine $\frac { d y } { d x }$ where $x ^ { 2 } y ^ { 3 } + x = y$ . Is $\frac { d y } { d x } = \frac { 2 x y ^ { 3 } + 1 } { - 3 x ^ { 2 } y ^ { 2 } + 1 }$ correct?

Accepted Answer

A) True 
 B)False

Question 3

Find the equation of the line tangent to the graph of $x ^ { 2 }$ $( y - 1 ) ^ { 3 }$ = 8 at the point (1, 3).
Enter your answer in slope-intercept form.

Accepted Answer

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

Question 4

When a manufacturer produces and sells x units per week, its weekly profit is P dollars, where $P = 100 \left( 2000 + 120 x - x ^ { 2 } \right)$ Production level t weeks from the present will be $x = \frac { t } { 2 } - 16$ Find the rate of change in profit with respect to x. Enter your answer as a polynomial in x in standard form (not labeled).

Accepted Answer

The answer of When a manufacturer produces and sells x...

Question 5

Differentiate
-y = $\frac { x + 1 } { x ^ { 2 } - 1 }$

Accepted Answer

A)  $\frac { x ^ { 3 } + x ^ { 2 } - x - 1 } { \left( x ^ { 2 } - 1 \right) ^ { 2 } }$ 
B)  $\frac { 1 } { 2 x }$ 
C)  - $\frac { 1 } { ( x - 1 ) ^ { 2 } }$ 
D)  none of these 
A)  $\frac { x ^ { 3 } + x ^ { 2 } - x - 1 } { \left( x ^ { 2 } - 1 \right) ^ { 2 } }$ 
B)  $\frac { 1 } { 2 x }$ 
C)  - $\frac { 1 } { ( x - 1 ) ^ { 2 } }$ 
D)  none of these

Question 6

Use implicit differentiation to determine the slope of the graph of $x ^ { 1 / 2 } + y ^ { 1 / 2 } = 4 \text { at } ( 1,9 )$ Enter just an integer.

Accepted Answer

The answer of Use implicit differentiation to determine the slope...

Question 7

Differentiate
-$x ^ { 5 }$ ∙ $\frac { x - 1 } { x + 1 }$ at x = 1
Enter just a reduced fraction.

Accepted Answer

The answer of Differentiate
-$x ^ { 5 }$ ∙ \(\frac...

Question 8

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

Accepted Answer

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

Question 9

Use implicit differentiation to determine $\frac { d y } { d x }$ where 4 $x ^ { 3 }$ + 4xy + y = 8. Is $\frac { d y } { d x } = \frac { - 12 x ^ { 2 } - 4 y } { 4 x + 1 }$ correct?

Accepted Answer

A) True 
 B)False

Question 10

Suppose 2 $x ^ { 3 }$ - 3 $p ^ { 4 }$ = 6, where x and p are differentiable functions of t. Find $\frac { \mathrm { dp } } { \mathrm { dt } }$ .

Accepted Answer

A)  6 $x ^ { 2 }$ $\frac { \mathrm { dx } } { \mathrm { dt } }$ - 12 $p ^ { 3 }$ 
B)  $\frac { 6 - 6 x ^ { 2 } } { 12 p ^ { 3 } }$ $\frac { \mathrm { dx } } { \mathrm { dt } }$ 
C)  6 $x ^ { 2 }$ - 12 $p ^ { 3 }$ $\frac { \mathrm { dp } } { \mathrm { dt } }$ 
D)  $\frac { x ^ { 2 } } { 2 p ^ { 3 } }$ $\frac { \mathrm { dx } } { \mathrm { dt } }$ 
A)  6 $x ^ { 2 }$ $\frac { \mathrm { dx } } { \mathrm { dt } }$ - 12 $p ^ { 3 }$ 
B)  $\frac { 6 - 6 x ^ { 2 } } { 12 p ^ { 3 } }$ $\frac { \mathrm { dx } } { \mathrm { dt } }$ 
C)  6 $x ^ { 2 }$ - 12 $p ^ { 3 }$ $\frac { \mathrm { dp } } { \mathrm { dt } }$ 
D)  $\frac { x ^ { 2 } } { 2 p ^ { 3 } }$ $\frac { \mathrm { dx } } { \mathrm { dt } }$

Question 11

The radius of a spherical balloon increases at a rate of 1 mm/sec. How fast is the surface area increasing when the radius is 10mm? (Note: The surface area S of a sphere of radius r is S = 4π $\mathrm { r } ^ { 2 }$ .) Enter just a real number (no approximations, no units or words).

Accepted Answer

The answer of The radius of a spherical balloon increases...

Question 12

Find the values of x where the tangent line is horizontal for the graph of $f ( x ) = \frac { 4 x ^ { 2 } } { x + 2 }$.

Accepted Answer

A)  x = 0, x = -4 
B)  x = 0, x = -2 
C)  x = -2 
D)  x = -2, x = 0, x = -4 
A)  x = 0, x = -4 
B)  x = 0, x = -2 
C)  x = -2 
D)  x = -2, x = 0, x = -4

Question 13

Suppose that the cost of manufacturing x units of a product is C(x) = 6x - 2 $\sqrt { x }$ + 1 dollars and that the production level t weeks from the present is x = 4 $t ^ { 2 }$ . Find the rate of change in cost with respect to time. Enter your answer as a polynomial in t in standard form (not labeled).

Accepted Answer

The answer of Suppose that the cost of manufacturing x...

Question 14

If f(x) = $\frac { 1 } { x ^ { 3 } + x ^ { 2 } }$ , find $f ^ { \prime }$ (x).

Accepted Answer

A)  $\frac { 1 } { \left( x ^ { 3 } + x ^ { 2 } \right) ^ { 2 } }$ 
B)  $\frac { x ^ { 3 } + x ^ { 2 } } { \left( x ^ { 3 } + x ^ { 2 } \right) ^ { 2 } }$ 
C)  $\frac { - 3 x ^ { 2 } - 2 x } { \left( x ^ { 3 } + x ^ { 2 } \right) ^ { 2 } }$ 
D)  - $\frac { 2 x + 1 } { x ^ { 2 } ( x + 1 ) ^ { 2 } }$ 
A)  $\frac { 1 } { \left( x ^ { 3 } + x ^ { 2 } \right) ^ { 2 } }$ 
B)  $\frac { x ^ { 3 } + x ^ { 2 } } { \left( x ^ { 3 } + x ^ { 2 } \right) ^ { 2 } }$ 
C)  $\frac { - 3 x ^ { 2 } - 2 x } { \left( x ^ { 3 } + x ^ { 2 } \right) ^ { 2 } }$ 
D)  - $\frac { 2 x + 1 } { x ^ { 2 } ( x + 1 ) ^ { 2 } }$

Question 15

Find the equation of the line tangent to the graph of $x ^ { 2 } y ^ { 3 } = 1 \text { at the point } ( 1,1 )$ Enter your answer in slope-intercept form.

Accepted Answer

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

Question 16

If f(x) = $\frac { 3 } { x }$ + $x ^ { 3 }$ and g(x) = 1 - $x ^ { 2 }$ , find $\frac { \mathrm { d } } { \mathrm { dx } }$ f(g(x)).

Accepted Answer

A)  $\frac { 6 x } { x ^ { 2 } }$ - 6 $x ^ { 2 }$ 
B)  - $\frac { 3 } { \left( 1 - x ^ { 2 } \right) ^ { 2 } }$ + 3 $\left( 1 - x ^ { 2 } \right) ^ { 2 }$ 
C)  $\frac { 6 x } { \left( 1 - x ^ { 2 } \right) ^ { 2 } }$ - 6x $\left( 1 - x ^ { 2 } \right) ^ { 2 }$ 
D)  $- \frac { 6 } { x } - 2 x ^ { 3 }$ $- \frac { 3 } { x ^ { 2 } } + 3 x ^ { 2 }$ 
A)  $\frac { 6 x } { x ^ { 2 } }$ - 6 $x ^ { 2 }$ 
B)  - $\frac { 3 } { \left( 1 - x ^ { 2 } \right) ^ { 2 } }$ + 3 $\left( 1 - x ^ { 2 } \right) ^ { 2 }$ 
C)  $\frac { 6 x } { \left( 1 - x ^ { 2 } \right) ^ { 2 } }$ - 6x $\left( 1 - x ^ { 2 } \right) ^ { 2 }$ 
D)  $- \frac { 6 } { x } - 2 x ^ { 3 }$ $- \frac { 3 } { x ^ { 2 } } + 3 x ^ { 2 }$

Question 17

Suppose that 15 $x ^ { 1 / 3 }$ $y ^ { - 2 / 3 }$ = 50, where x and y are both differentiable functions of t. Find $\frac { d y } { d t }$ .

Accepted Answer

A)  $\frac { y } { 2 x }$ $\frac { \mathrm { dx } } { \mathrm { dt } }$ 
B)  $\frac { 10 } { x ^ { 2 / 3 } y 5 / 3 }$ $\frac { \mathrm { dx } } { \mathrm { dt } }$ 
C)  $\frac { - y } { x }$ $\frac { \mathrm { dx } } { \mathrm { dt } }$ 
D)  $\frac { 25 y ^ { 1 / 3 } } { 2 x ^ { 2 / 3 } }$ 
A)  $\frac { y } { 2 x }$ $\frac { \mathrm { dx } } { \mathrm { dt } }$ 
B)  $\frac { 10 } { x ^ { 2 / 3 } y 5 / 3 }$ $\frac { \mathrm { dx } } { \mathrm { dt } }$ 
C)  $\frac { - y } { x }$ $\frac { \mathrm { dx } } { \mathrm { dt } }$ 
D)  $\frac { 25 y ^ { 1 / 3 } } { 2 x ^ { 2 / 3 } }$

Question 18

Differentiate
-y = $x ^ { 2 }$ + 4x( $x ^ { 3 / 2 }$ )

Accepted Answer

A)  2x + 10 $x ^ { 3 / 2 }$ 
B)  $\frac { x ^ { 3 / 2 } ( 2 x + 4 ) - \left( x ^ { 2 } + 4 x \right) \left( \frac { 3 } { 2 x ^ { 1 / 2 } } \right) } { \left( x ^ { 3 / 2 } \right) x ^ { 2 } }$ 
C)  ( $x ^ { 2 }$ + 4x) $\left( \frac { 3 } { 2 x ^ { 1 / 2 } } \right.$ + ( $x ^ { 3 / 2 }$ )(2x + 4) 
D)  6x( $x ^ { 1 / 2 }$ ) + 4 $x ^ { 3 / 2 }$ 
A)  2x + 10 $x ^ { 3 / 2 }$ 
B)  $\frac { x ^ { 3 / 2 } ( 2 x + 4 ) - \left( x ^ { 2 } + 4 x \right) \left( \frac { 3 } { 2 x ^ { 1 / 2 } } \right) } { \left( x ^ { 3 / 2 } \right) x ^ { 2 } }$ 
C)  ( $x ^ { 2 }$ + 4x) $\left( \frac { 3 } { 2 x ^ { 1 / 2 } } \right.$ + ( $x ^ { 3 / 2 }$ )(2x + 4) 
D)  6x( $x ^ { 1 / 2 }$ ) + 4 $x ^ { 3 / 2 }$

Question 19

The following function may be viewed as a composite function f(g(x)). Find f(x) and g(x). $\left( 3 x ^ { 2 } - 2 x + 1 \right) ^ { 6 }$

Accepted Answer

A)  f(x) = x, G(x) = 3 $x ^ { 2 }$ - 2x + 1 
B)  f(x) = $x ^ { 6 }$, G(x) = 3 $x ^ { 2 }$ - 2x + 1 
C)  f(x) = 3 $x ^ { 2 }$ - 2x + 1, G(x) = $x ^ { 6 }$ 
D)  f(x) = $x ^ { 6 }$, G(x) = $\left( 3 x ^ { 2 } - 2 x + 1 \right) ^ { 6 }$ 
A)  f(x) = x, G(x) = 3 $x ^ { 2 }$ - 2x + 1 
B)  f(x) = $x ^ { 6 }$, G(x) = 3 $x ^ { 2 }$ - 2x + 1 
C)  f(x) = 3 $x ^ { 2 }$ - 2x + 1, G(x) = $x ^ { 6 }$ 
D)  f(x) = $x ^ { 6 }$, G(x) = $\left( 3 x ^ { 2 } - 2 x + 1 \right) ^ { 6 }$

Question 20

Differentiate
-f(x) = $\frac { x ^ { 2 } } { x ^ { 3 } - 5 x + 2 }$ at x = -1
Enter a reduced fraction only.

Accepted Answer

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

Use implicit differentiation to determine the slope of the graph of $\sqrt { x + y }$ = x at (2, 2). Enter just an integer.

Use implicit differentiation to determine $\frac { d y } { d x }$ where $x ^ { 2 } y ^ { 3 } + x = y$ . Is $\frac { d y } { d x } = \frac { 2 x y ^ { 3 } + 1 } { - 3 x ^ { 2 } y ^ { 2 } + 1 }$ correct?

Find the equation of the line tangent to the graph of $x ^ { 2 }$ $( y - 1 ) ^ { 3 }$ = 8 at the point (1, 3). Enter your answer in slope-intercept form.

Differentiate -y = $\frac { x + 1 } { x ^ { 2 } - 1 }$

Use implicit differentiation to determine the slope of the graph of $x ^ { 1 / 2 } + y ^ { 1 / 2 } = 4 \text { at } ( 1,9 )$ Enter just an integer.

Differentiate - $x ^ { 5 }$ ∙ $\frac { x - 1 } { x + 1 }$ at x = 1 Enter just a reduced fraction.

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

Use implicit differentiation to determine $\frac { d y } { d x }$ where 4 $x ^ { 3 }$ + 4xy + y = 8. Is $\frac { d y } { d x } = \frac { - 12 x ^ { 2 } - 4 y } { 4 x + 1 }$ correct?

Suppose 2 $x ^ { 3 }$ - 3 $p ^ { 4 }$ = 6, where x and p are differentiable functions of t. Find $\frac { \mathrm { dp } } { \mathrm { dt } }$ .

The radius of a spherical balloon increases at a rate of 1 mm/sec. How fast is the surface area increasing when the radius is 10mm? (Note: The surface area S of a sphere of radius r is S = 4π $\mathrm { r } ^ { 2 }$ .) Enter just a real number (no approximations, no units or words).

Find the values of x where the tangent line is horizontal for the graph of $f ( x ) = \frac { 4 x ^ { 2 } } { x + 2 }$ .

If f(x) = $\frac { 1 } { x ^ { 3 } + x ^ { 2 } }$ , find $f ^ { \prime }$ (x).

Find the equation of the line tangent to the graph of $x ^ { 2 } y ^ { 3 } = 1 \text { at the point } ( 1,1 )$ Enter your answer in slope-intercept form.

If f(x) = $\frac { 3 } { x }$ + $x ^ { 3 }$ and g(x) = 1 - $x ^ { 2 }$ , find $\frac { \mathrm { d } } { \mathrm { dx } }$ f(g(x)).

Suppose that 15 $x ^ { 1 / 3 }$ $y ^ { - 2 / 3 }$ = 50, where x and y are both differentiable functions of t. Find $\frac { d y } { d t }$ .

Differentiate -y = $x ^ { 2 }$ + 4x( $x ^ { 3 / 2 }$ )

The following function may be viewed as a composite function f(g(x)). Find f(x) and g(x). $\left( 3 x ^ { 2 } - 2 x + 1 \right) ^ { 6 }$

Differentiate -f(x) = $\frac { x ^ { 2 } } { x ^ { 3 } - 5 x + 2 }$ at x = -1 Enter a reduced fraction only.

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

Use implicit differentiation to determine the slope of the graph of x+y\sqrt { x + y }x+y​ = x at (2, 2). Enter just an integer.

Use implicit differentiation to determine dydx\frac { d y } { d x }dxdy​ where x2y3+x=yx ^ { 2 } y ^ { 3 } + x = yx2y3+x=y . Is dydx=2xy3+1−3x2y2+1\frac { d y } { d x } = \frac { 2 x y ^ { 3 } + 1 } { - 3 x ^ { 2 } y ^ { 2 } + 1 }dxdy​=−3x2y2+12xy3+1​ correct?

Find the equation of the line tangent to the graph of x2x ^ { 2 }x2 (y−1)3( y - 1 ) ^ { 3 }(y−1)3 = 8 at the point (1, 3). Enter your answer in slope-intercept form.

Differentiate -y = x+1x2−1\frac { x + 1 } { x ^ { 2 } - 1 }x2−1x+1​

Use implicit differentiation to determine the slope of the graph of x1/2+y1/2=4 at (1,9)x ^ { 1 / 2 } + y ^ { 1 / 2 } = 4 \text { at } ( 1,9 )x1/2+y1/2=4 at (1,9) Enter just an integer.

Differentiate - x5x ^ { 5 }x5 ∙ x−1x+1\frac { x - 1 } { x + 1 }x+1x−1​ at x = 1 Enter just a reduced fraction.

Differentiate -f(x) = ( x2x ^ { 2 }x2 + x)(3 x2x ^ { 2 }x2 + 4x - 1) at x = 1 Enter just an integer.

Use implicit differentiation to determine dydx\frac { d y } { d x }dxdy​ where 4 x3x ^ { 3 }x3 + 4xy + y = 8. Is dydx=−12x2−4y4x+1\frac { d y } { d x } = \frac { - 12 x ^ { 2 } - 4 y } { 4 x + 1 }dxdy​=4x+1−12x2−4y​ correct?

Suppose 2 x3x ^ { 3 }x3 - 3 p4p ^ { 4 }p4 = 6, where x and p are differentiable functions of t. Find dpdt\frac { \mathrm { dp } } { \mathrm { dt } }dtdp​ .

The radius of a spherical balloon increases at a rate of 1 mm/sec. How fast is the surface area increasing when the radius is 10mm? (Note: The surface area S of a sphere of radius r is S = 4π r2\mathrm { r } ^ { 2 }r2 .) Enter just a real number (no approximations, no units or words).

Find the values of x where the tangent line is horizontal for the graph of f(x)=4x2x+2f ( x ) = \frac { 4 x ^ { 2 } } { x + 2 }f(x)=x+24x2​ .

If f(x) = 1x3+x2\frac { 1 } { x ^ { 3 } + x ^ { 2 } }x3+x21​ , find f′f ^ { \prime }f′ (x).

Find the equation of the line tangent to the graph of x2y3=1 at the point (1,1)x ^ { 2 } y ^ { 3 } = 1 \text { at the point } ( 1,1 )x2y3=1 at the point (1,1) Enter your answer in slope-intercept form.

If f(x) = 3x\frac { 3 } { x }x3​ + x3x ^ { 3 }x3 and g(x) = 1 - x2x ^ { 2 }x2 , find ddx\frac { \mathrm { d } } { \mathrm { dx } }dxd​ f(g(x)).

Suppose that 15 x1/3x ^ { 1 / 3 }x1/3 y−2/3y ^ { - 2 / 3 }y−2/3 = 50, where x and y are both differentiable functions of t. Find dydt\frac { d y } { d t }dtdy​ .

Differentiate -y = x2x ^ { 2 }x2 + 4x( x3/2x ^ { 3 / 2 }x3/2 )

The following function may be viewed as a composite function f(g(x)). Find f(x) and g(x). (3x2−2x+1)6\left( 3 x ^ { 2 } - 2 x + 1 \right) ^ { 6 }(3x2−2x+1)6

Differentiate -f(x) = x2x3−5x+2\frac { x ^ { 2 } } { x ^ { 3 } - 5 x + 2 }x3−5x+2x2​ at x = -1 Enter a reduced fraction only.

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

Use implicit differentiation to determine the slope of the graph of $\sqrt { x + y }$ = x at (2, 2). Enter just an integer.

Use implicit differentiation to determine $\frac { d y } { d x }$ where $x ^ { 2 } y ^ { 3 } + x = y$ . Is $\frac { d y } { d x } = \frac { 2 x y ^ { 3 } + 1 } { - 3 x ^ { 2 } y ^ { 2 } + 1 }$ correct?

Find the equation of the line tangent to the graph of $x ^ { 2 }$ $( y - 1 ) ^ { 3 }$ = 8 at the point (1, 3). Enter your answer in slope-intercept form.

Differentiate -y = $\frac { x + 1 } { x ^ { 2 } - 1 }$

Use implicit differentiation to determine the slope of the graph of $x ^ { 1 / 2 } + y ^ { 1 / 2 } = 4 \text { at } ( 1,9 )$ Enter just an integer.

Differentiate - $x ^ { 5 }$ ∙ $\frac { x - 1 } { x + 1 }$ at x = 1 Enter just a reduced fraction.

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

Use implicit differentiation to determine $\frac { d y } { d x }$ where 4 $x ^ { 3 }$ + 4xy + y = 8. Is $\frac { d y } { d x } = \frac { - 12 x ^ { 2 } - 4 y } { 4 x + 1 }$ correct?

Suppose 2 $x ^ { 3 }$ - 3 $p ^ { 4 }$ = 6, where x and p are differentiable functions of t. Find $\frac { \mathrm { dp } } { \mathrm { dt } }$ .

The radius of a spherical balloon increases at a rate of 1 mm/sec. How fast is the surface area increasing when the radius is 10mm? (Note: The surface area S of a sphere of radius r is S = 4π $\mathrm { r } ^ { 2 }$ .) Enter just a real number (no approximations, no units or words).

Find the values of x where the tangent line is horizontal for the graph of $f ( x ) = \frac { 4 x ^ { 2 } } { x + 2 }$ .

If f(x) = $\frac { 1 } { x ^ { 3 } + x ^ { 2 } }$ , find $f ^ { \prime }$ (x).

Find the equation of the line tangent to the graph of $x ^ { 2 } y ^ { 3 } = 1 \text { at the point } ( 1,1 )$ Enter your answer in slope-intercept form.

If f(x) = $\frac { 3 } { x }$ + $x ^ { 3 }$ and g(x) = 1 - $x ^ { 2 }$ , find $\frac { \mathrm { d } } { \mathrm { dx } }$ f(g(x)).

Suppose that 15 $x ^ { 1 / 3 }$ $y ^ { - 2 / 3 }$ = 50, where x and y are both differentiable functions of t. Find $\frac { d y } { d t }$ .

Differentiate -y = $x ^ { 2 }$ + 4x( $x ^ { 3 / 2 }$ )

The following function may be viewed as a composite function f(g(x)). Find f(x) and g(x). $\left( 3 x ^ { 2 } - 2 x + 1 \right) ^ { 6 }$

Differentiate -f(x) = $\frac { x ^ { 2 } } { x ^ { 3 } - 5 x + 2 }$ at x = -1 Enter a reduced fraction only.