Deck 3: Techniques of Differentiation

Question 1

Find the slope of the tangent line to f(x) = at (-1, f(-1)).
Enter a reduced fraction.

Answer 1

Question 2

Differentiate
y = + 4x( )

A) 2x + 10
B)
C) ( + 4x) + ( )(2x + 4)
D) 6x( ) + 4

Answer 2

2x + 10

Question 3

Differentiate
f(x) = at x = -1
Enter just a reduced fraction.

Answer 3

Question 4

Find the slope of the tangent line to f(x) = at (2, f(2)).
Enter just a reduced fraction.

Question 5

Differentiate
f(x) = at x = 2
Enter just a reduced fraction.

Question 6

Differentiate
y =

A)
B)
C) -
D) none of these

Question 7

Differentiate
f(x) = at x = -1
Enter a reduced fraction only.

Question 8

Find the slope of the tangent line to f(x) = 4( + 1)(2 + 2x + 1 at (-1, f(-1)).
Enter just an integer.

Question 9

Differentiate
f(x) =

A) -
B)
C) -
D)

Question 10

Find the slope of the tangent line to f(x) = at (-1, f(-1)).
Enter just an integer.

Question 11

Differentiate
f(x) = ( + 1 ( - 1 at x = -1
Enter just an integer.

Question 12

Differentiate
f(x) = (2x + 1) at x = 1
Enter just an integer.

Question 13

Differentiate
f(x) = ( + x)(3 + 4x - 1) at x = 1
Enter just an integer.

Question 14

Differentiate
f(x) = (4 + 4)(2 + 2x) at x = 1
Enter just an integer.

Question 15

Differentiate
f(x) = ( - 2 ( + 2 at x = -1
Enter just an integer.

Question 16

Differentiate

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

Differentiate
g(x) = ( + 1)(3 - 1)

A) -3 + 3 + 6x
B) 15 - 3 + 6x
C) -3 - 3 + 6x
D) 3 - 3 - 6x

Question 18

Differentiate
∙ at x = 1
Enter just a reduced fraction.

Question 19

Differentiate
f(x) = ( - 8 + 5)( + - 1) at x = 1
Enter just an integer.

Question 20

Differentiate
F(r) = (r - 1)(r + 1

A) (2 + 2 - 2r)(r + 1
B) 2r( + r -1)
C)
D) 4 - 2r

Question 21

The following function may be viewed as a composite function f(g(x)). Find f(x) and g(x).

A) f(x) = x, G(x) = + 2x - 3
B) f(x) = , G(x) =
C) f(x) = , G(x) = + 2x - 3
D) f(x) = + 2x - 3, G(x) =

Question 22

If f(x) = - 9 and g(x) = - 16, find g(f(x)).

A) ( - 9 - 16
B) - 36x
C) (( - 9 - 16)(2x)
D) 4 - 50

Question 23

Let f(x) = . Using the chain rule, find an expression for the derivative of [f(g(x))].

A) 3 g'(x
B) 3[g(x)
C) g'(x
D) 3[g(x) g'(x)
E) none of these

Question 24

One hour after x milligrams of a particular drug are given to a person, the change in body temperature T(x), in degrees Celsius, is given approximately by: T(x) = - , 0 ≤ x ≤ 6. Find the sensitivity, T'(x), of the body to a dosage of three milligrams.

A) degrees per milligram
B) - degrees per milligram
C) - degrees per milligram
D) - degree per milligram

Question 25

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

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 26

Find the values of x where the tangent line is horizontal for the graph of

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

Question 27

Find the equation of the tangent line to the graph of the function at

A) y = x
B) y = x -
C) y = -3x + 4
D) y = 3x - 2

Question 28

The following function may be viewed as a composite function f(g(x)). Find f(x) and g(x).

A) f(x) = x, G(x) = 3 - 2x + 1
B) f(x) = , G(x) = 3 - 2x + 1
C) f(x) = 3 - 2x + 1, G(x) =
D) f(x) = , G(x) =

Question 29

Use the chain rule to compute the derivative of at x = -1.
Enter just a reduced fraction.

Question 30

If y = ( + 1 , find .

A) ( + 1
B) ( + 1
C) 5( + 1
D) 5( + 1
E) none of these

Question 31

Let g(x) = . Using the chain rule, find an expression for the derivative of [g(f(x))].

A) f'(x)
B) f'( )
C) f'(x) + f(x)
D) f( )
E) none of these

Question 32

Let f(x) = , g(x) = . Compute f(g(x)) at x= 4.
Enter just a reduced fraction.

Question 33

Find all x-coordinates of points (x, y) on the graph of Q(x) = where the tangent line is horizontal. Enter your answer as just a, or a, b where these are integers and < b.

Question 34

The radius, r, of a sphere is increasing. For what value of r is equal to 64π times the rate of increase of r. Enter just an integer.

Question 35

Find the slope of the tangent line to f(x) = at (1, f(1)).
Enter just a reduced fraction of form .

Question 36

If f(x) = + and g(x) = 1 - , find f(g(x)).

A) - 6
B) - + 3
C) - 6x
D)

Question 37

Use the chain rule to find the derivative of at x = 1.
Enter just an integer.

Question 38

Differentiate: f(x) = (( + 2 + 1 at x = -1.
Enter just an integer.

Question 39

If f(x) = , find (x).

A)
B)
C)
D) -

Question 40

Find the slope of the tangent line to the graph of y = at the point (2, 1).
Enter just an integer.

Question 41

If y = 3u + 2 and u = , find .

A)
B)
C)
D)

Question 42

Use implicit differentiation to determine the slope of the graph of Enter just an integer.

Question 43

Suppose that x and y are related by the equation + = . Use implicit differentiation to determine .

A) =
B) = 3 + 2(y + 1)
C) =
D) =

Question 44

Use implicit differentiation to determine where 4 + 4xy + y = 8. Is correct?

Question 45

If y = and u = - 5 + 1, find .

A)
B) - + 1)
C)
D)

Question 46

Use implicit differentiation to determine the slope of the graph of = x at (2, 2).
Enter just an integer.

Question 47

Use implicit differentiation to determine where + = 2xy. Is correct?

Question 48

Use implicit differentiation to determine where + = 4. Is correct?

Question 49

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

Question 50

Compute using the chain rule where and u = 3x + 4.
Enter your answer as just a polynomial in x in standard form (no label).

Question 51

Compute y = 4 + 8u + 4 and u = 3x + 1.
Enter your answer as just a polynomial in x in standard form (no label).

Question 52

Use implicit differentiation to determine where . Is correct?

Question 53

Suppose that x and y are related by the equation + = 4. Use implicit differentiation to determine .

A) = -
B) =
C) = , y ≠ 0
D) = - , y ≠ 0

Question 54

Compute using the chain rule where y = and u = at x = 1.
Enter just a reduced fraction.

Question 55

If y = u + 1 - and u = + 1, find .

A)
B) - ∙
C) -
D) -2 ∙

Question 56

If f(x) = x and g(x) = , find f(g(x)).

A) 2x + 10
B) + 5
C) 2x + 10 ( - 1)
D) 2x + 10

Question 57

Use implicit differentiation to determine where xy + 10 = . Is correct?

Question 58

Assume 4 + 2xy - = . What is the slope of the graph at the point ?

A) -
B) 8
C)
D) 1

Question 59

The cost of manufacturing x units is C dollars, where C = 4x + 6 + 5. Weekly production at t weeks from the present is estimated to be x = 2800 + 100t units when t = 8. Find the time rate of change of cost, . Enter just an integer.

Question 60

When a manufacturer produces and sells x units per week, its weekly profit is P dollars, where Production level t weeks from the present will be Find the rate of change in profit with respect to x. Enter your answer as a polynomial in x in standard form (not labeled).

Question 61

Suppose 2 - 3 = 6, where x and p are differentiable functions of t. Find .

A) 6 - 12
B)
C) 6 - 12
D)

Question 62

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

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 63

Mr. Smith is 6 ft tall and walks at a constant rate of 2 ft/sec toward a street light that is 10 ft above the ground. At what rate is the length of his shadow changing when he is 6 ft from the base of the pole that supports the light? Enter just an integer (no units).

Question 64

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 .) Enter just a real number (no approximations, no units or words).

Question 65

Assume + = x. What is the slope of the graph at the point (-1, 9)?

A) 30
B) 4 +
C) 3
D) 18

Question 66

Find the equation of the tangent line to the graph of + = 1 at the point .
Is correct?

Question 67

Find the equation of the line tangent to the graph of Enter your answer in slope-intercept form.

Question 68

Find the equation of the line tangent to the graph of = 8 at the point (1, 3).
Enter your answer in slope-intercept form.

Question 69

Determine the rate of change of with respect to x at x = 1.
Enter just a reduced quotient of form .