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04 The Derivative Online Quiz Questions

Use this free practice quiz with 20 questions to review 04 The Derivative, test your knowledge, and prepare for your next test or exam.

20 questions
01
True or false
1 point

True or false: The standard formulas for derivatives of trigonometric functions assume that angles are measured in radians.

  1. A

    True

  2. B

    False

02
Choose one
1 point

Let f(x)=ln(3x+1). What is f'(1)?

  1. A

    1/4

  2. B

    1/2

  3. C

    3/4

  4. D

    3

03
True or false
1 point

True or false: If f'(a)=0, then f has a local maximum or local minimum at x=a.

  1. A

    True

  2. B

    False

04
Choose all
1 point

Select all statements that are correct about differentiability and continuity.

  1. A

    If a function is differentiable at a point, it is continuous there.

  2. B

    A continuous function need not be differentiable at every point.

  3. C

    The function f(x)=|x| is continuous at 0 but not differentiable there.

  4. D

    Every continuous function has a well-defined finite tangent slope at every point.

05
Choose one
1 point

Which expression is the derivative of x^2 sin x?

  1. A

    2x cos x + x^2 sin x

  2. B

    x^2 cos x

  3. C

    2x sin x + x^2 cos x

  4. D

    2x sin x + x^2 sin x

06
Written response
1 point

For f(x)=4x^5-3x^2+7x-9, enter the exact value of f'(1).

07
Written response
1 point

What is the standard term for [f(a+h)-f(a)]/h?

08
Choose all
1 point

Select all correct derivative statements.

  1. A

    The derivative of e^(2x) is 2e^(2x).

  2. B

    The derivative of ln(x^2+1) is 2x/(x^2+1).

  3. C

    The derivative of sin(5x) is 5 cos(5x).

  4. D

    The derivative of cos x is sin x.

09
Fill in the blank
1 point

Complete the chain rule: the derivative of f(g(x)) is multiplied by .

10
Fill in the blank
1 point

For f(x)=x^2 at x=3, complete the tangent-line equation y-9=.

11
Open ended
1 point

Use the limit definition of the derivative to derive the derivative function for f(x)=x^2. Show the main algebraic steps and state the final result.

12
Choose one
1 point

Which expression is the derivative of cos(4x^3-1)?

  1. A

    12x^2 sin(4x^3-1)

  2. B

    -12x^2 sin(4x^3-1)

  3. C

    -4x^3 sin(4x^3-1)

  4. D

    -12x^2 cos(4x^3-1)

13
Choose one
1 point

What does the derivative of a function at a point represent geometrically?

  1. A

    The total area under the curve up to that point

  2. B

    The slope of the tangent line at that point

  3. C

    The average height of the function over an interval

  4. D

    The value of the function's second derivative at that point

14
True or false
1 point

If a function is differentiable at a point, then it must be continuous at that point.

  1. A

    True

  2. B

    False

15
Choose one
1 point

For f(x)=x2f(x)=x^2, what is the instantaneous rate of change at x=3x=3?

  1. A

    2

  2. B

    3

  3. C

    6

  4. D

    9

16
Choose one
1 point

What is the derivative of f(x)=4x5−3x2+7x−9f(x)=4x^5-3x^2+7x-9?

  1. A

    20x4−6x2+720x^4-6x^2+7

  2. B

    4x4−6x+74x^4-6x+7

  3. C

    20x4−6x+720x^4-6x+7

  4. D

    20x5−6x2+7x−920x^5-6x^2+7x-9

17
Choose one
1 point

Which equation is the tangent line to f(x)=x2f(x)=x^2 at x=3x=3?

  1. A

    y=3x+9y=3x+9

  2. B

    y=6x−9y=6x-9

  3. C

    y=6x+9y=6x+9

  4. D

    y=9x−6y=9x-6

18
Written response
1 point

For f(x)=ln⁡(x2+1)f(x)=\ln(x^2+1), enter the numerical value of f′(1)f'(1).

19
Choose one
1 point

What is ddx[(3x2+1)5]\frac{d}{dx}[(3x^2+1)^5]?

  1. A

    5(3x2+1)45(3x^2+1)^4

  2. B

    30(3x2+1)430(3x^2+1)^4

  3. C

    15x(3x2+1)515x(3x^2+1)^5

  4. D

    30x(3x2+1)430x(3x^2+1)^4

20
Written response
1 point

A cost function C(q) measures total cost in dollars when q items are produced. What units does C'(q) have?