Master Calculus Derivatives: Online Quizzes and Practice

Question 1

Find the derivative of the function f(x) = (sin(x) + cos(x))^2.

Accepted Answer

2(sin(x) + cos(x))(cos(x) - sin(x))

Answer

2(sin(x) - cos(x))(cos(x) + sin(x))

Answer

4(sin(x) + cos(x))

Question 2

Which of the following is the derivative of f(x) = e^x?

Accepted Answer

e^x

Answer

x*e^x

Answer

e^x/x

Answer

x/e^x

Question 3

Find the derivative of the function f(x) = ln(x).

Accepted Answer

1/x

Answer

x

Answer

-1/x

Answer

-x

Question 4

Use the product rule to find the derivative of the function f(x) = (x+1)(x-2).

Accepted Answer

x^2 - x - 2

Answer

2x-1

Answer

x^2 + x - 2

Answer

2x + 1

Question 5

Which of the following is the derivative of the function f(x) = sqrt(x)?

Accepted Answer

1/2sqrt(x)

Answer

2sqrt(x)

Answer

-1/2sqrt(x)

Answer

-2sqrt(x)

Question 6

Find the derivative of the function f(x) = x^2 + 2x - 1.

Accepted Answer

2x + 2

Answer

x + 2

Answer

2x - 2

Answer

x - 2

Question 7

Identify the critical point(s) of the function f(x) = x^3 - 6x^2 + 9x.

Accepted Answer

x = 0, 3

Answer

x = 1, 3

Answer

x = 0, 2

Answer

x = -1, 3

Question 8

Find the derivative of the function f(x) = 2x^3 + 5x^2 - 7x + 1.

Accepted Answer

6x^2 + 10x - 7

Answer

3x^2 + 10x - 7

Answer

2x^3 + 10x^2 - 7

Answer

3x^2 + 5x - 7

Question 9

Determine the derivative of the function f(x) = sin(x).

Accepted Answer

cos(x)

Answer

-cos(x)

Answer

sin(2x)

Answer

-sin(2x)

Question 10

Find the derivative of the function f(x) = e^x.

Accepted Answer

e^x

Answer

x * e^x

Answer

2 * e^x

Answer

e^(x+1)

Question 11

Determine the derivative of the function f(x) = ln(x).

Accepted Answer

1/x

Answer

-1/x

Answer

x

Answer

-x

Question 12

Find the derivative of the function f(x) = x^3 - 2x^2 + 5x - 1.

Accepted Answer

3x^2 - 4x + 5

Answer

x^2 - 4x + 5

Answer

3x - 4

Answer

x - 4

Question 13

What is the derivative of the function f(x) = sin(x)?

Accepted Answer

cos(x)

Answer

-cos(x)

Answer

sin(x)

Answer

-sin(x)

Question 14

Find the critical points of the function f(x) = x^2 - 4x + 3.

Accepted Answer

x = 2

Answer

x = -2

Answer

x = 1

Answer

The function has no critical points

Question 15

Which of the following is an application of derivatives in optimization?

Accepted Answer

Finding the maximum or minimum value of a function

Answer

Calculating the instantaneous velocity of an object

Answer

Determining the slope of a line

Answer

Integrating functions

Question 16

Find the derivative of the function f(x) = ln(x).

Accepted Answer

1/x

Answer

x

Answer

ln(x)

Answer

1

Question 17

What is the chain rule of differentiation?

Accepted Answer

If f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x)

Answer

If f(x) = g(x) * h(x), then f'(x) = g'(x) + h'(x)

Answer

If f(x) = g(x) / h(x), then f'(x) = g'(x) - h'(x)

Answer

If f(x) = g(x) ^ h(x), then f'(x) = g'(x) * h'(x) / h(x)

Question 18

Find the derivative of the function f(x) = e^(2x).

Accepted Answer

2e^(2x)

Answer

e^(2x)

Answer

2x * e^(2x)

Answer

x * e^(2x)

Question 19

What is the relationship between the first derivative of a function and its graph?

Accepted Answer

The first derivative is the slope of the graph.

Answer

The first derivative is the area under the graph.

Answer

The first derivative is the inverse of the function.

Answer

The first derivative is the integral of the function.

Question 20

Find the derivative of the function g(x) = sin(2x).

Accepted Answer

2cos(2x)

Answer

cos(2x)

Answer

-2sin(2x)

Answer

sin(2x)

Question 21

Which of the following is NOT an application of derivatives?

Accepted Answer

Graphing functions

Answer

Finding rates of change

Answer

Solving optimization problems

Answer

Checking for concavity

Question 22

Find the derivative of f(x) = e^(2x + 1).

Accepted Answer

2e^(2x + 1)

Answer

e^(2x + 1)

Answer

2xe^(2x + 1)

Answer

xe^(2x + 1)

Question 23

Which of the following is the second derivative of f(x) = x^4 + 2x^2 - 1?

Accepted Answer

12x + 4

Answer

12x - 4

Answer

4x + 12

Answer

4x - 12

Question 24

Given f(x) = (x^2 + 1)^3, find f'(x).

Accepted Answer

3(x^2 + 1)^2(2x)

Answer

3x(x^2 + 1)^2

Answer

6x(x^2 + 1)^2

Answer

2x(x^2 + 1)^3

Question 25

Determine the derivative of y = sin(2x).

Accepted Answer

2cos(2x)

Answer

-2cos(2x)

Answer

sin(2x)

Answer

2sin(2x)

Question 26

Find the derivative of y = (2x + 1)(x^2 - 3).

Accepted Answer

2x^3 - 5x^2 - 3x + 2

Answer

2x^3 - 5x^2 + 3x + 2

Answer

4x^3 - 5x^2 - 3x + 2

Answer

4x^3 - 5x^2 + 3x + 2

Question 27

Find the derivative of y = e^(3x - 2).

Accepted Answer

3e^(3x - 2)

Answer

e^(3x - 2)

Answer

3e^(3)

Answer

e^3

Question 28

Find the derivative of the function f(x) = ln(x² + 1).

Accepted Answer

2x / (x² + 1)

Answer

1 / (x² + 1)

Answer

x / (x² + 1)

Answer

(x² + 1) / x

Question 29

Find the derivative of the function f(x) = sin(x) + cos(x).

Accepted Answer

cos(x) + sin(x)

Answer

cos(x) - sin(x)

Answer

-sin(x) - cos(x)

Answer

-cos(x) + sin(x)

Question 30

Find the derivative of the function f(x) = (x² + 1) / (x - 1).

Accepted Answer

(2x - 1) / (x - 1)^2

Answer

(x² + 1) / (x - 1)

Answer

(x - 1) / (x² + 1)

Answer

(x² + 1) * (x - 1)

Question 31

Find the critical points of the function f(x) = x³ - 3x² + 2.

Accepted Answer

x = 0, x = 2

Answer

x = 0, x = 1, x = 2

Answer

x = 1, x = 2

Answer

x = 0, x = 1

Question 32

Which of the following is NOT a method used in differentiation?

Accepted Answer

Substitution

Answer

Chain rule

Answer

Quotient rule

Answer

Product rule

Question 33

Find the derivative of f(x) = e^(2x):

Accepted Answer

2e^(2x)

Answer

2x*e^(2x)

Answer

e^(4x)

Answer

(1/2)*e^(2x)

Question 34

Calculate the derivative of y = sin(3x):

Accepted Answer

3cos(3x)

Answer

-3sin(3x)

Answer

cos(3x)

Answer

sin(3x)

Question 35

What is the value of the derivative of f(x) = |x| at x = 0?

Accepted Answer

Does not exist

Answer

1

Answer

0

Question 36

Which of the following is an application of derivatives in optimization?

Accepted Answer

Finding the minimum or maximum value of a function

Answer

Calculating areas under curves

Answer

Solving differential equations

Answer

Evaluating limits

Question 37

Find the derivative of the function f(x) = e^(4x + 1).

Accepted Answer

4e^(4x + 1)

Answer

-e^(4x + 1) / 4

Answer

4x * e^(4x + 1)

Answer

e^(4x + 1) / 4

Question 38

Calculate the derivative of y = ln(x^3 + 2).

Accepted Answer

3x / (x^3 + 2)

Answer

1 / (x^3 + 2)

Answer

3 / (x^3 + 2)

Answer

x / (x^3 + 2)

Question 39

Find the derivative of the implicit function defined by x² + y³ = 5.

Accepted Answer

dy/dx = -2x / 3y^2

Answer

-3y^2 / 2x

Answer

2x / 3y^2

Answer

3y^2 / 2x

Question 40

Determine the derivative of f(x) = |x + 2|.

Accepted Answer

1 if x > -2, -1 if x < -2, undefined if x = -2

Answer

1

Answer

-1

Question 41

Which of the following is the definition of the derivative of a function f(x) at a point x?

Accepted Answer

The limit of the difference quotient as h approaches 0

Answer

The slope of the tangent line to the graph of f(x) at x

Answer

The value of f(x) at x

Question 42

Find the derivative of the function f(x) = x^2 + 2x - 1.

Accepted Answer

2x + 2

Answer

1

Answer

x^2 + 2x

Answer

2x - 1

Question 43

What is the derivative of the function f(x) = sin(x)?

Accepted Answer

cos(x)

Answer

sin(x)

Answer

-cos(x)

Answer

-sin(x)

Question 44

Find the derivative of the function f(x) = e^x.

Accepted Answer

e^x

Answer

e^(-x)

Answer

1

Answer

x * e^x

Question 45

What is the first derivative of the function f(x) = ln(x)?

Accepted Answer

1/x

Answer

x

Answer

-1/x

Answer

ln(x)

Question 46

Find the second derivative of the function f(x) = x^3.

Accepted Answer

6x

Answer

x^2

Answer

1

Answer

3x^2

Question 47

Use the Chain Rule to find the derivative of the function f(x) = (x^2 + 1)^3.

Accepted Answer

6x(x^2 + 2)

Answer

2x(x^2 + 1)

Answer

(x^2 + 1)^2

Answer

3(x^2 + 2)(2x)

Question 48

A particle moves along a straight line with its position given by s(t) = t^3 - 6t^2 + 9t. What is the particle's instantaneous velocity at t = 2?

Accepted Answer

0

Answer

1

Answer

-2

Answer

2

Question 49

Find the derivative of f(x) = 2x^3 - 5x^2 + 3x - 7.

Accepted Answer

6x^2 - 10x + 3

Answer

2x^2 - 5x + 3

Answer

12x^2 - 10x - 3

Answer

6x^3 - 10x - 7

Question 50

Find the slope of the tangent line to the graph of f(x) = x^2 + 2x - 3 at x = 1.

Accepted Answer

4

Answer

-2

Answer

6

Answer

2

Question 51

Find the critical points of f(x) = x^3 - 3x^2 + 2x + 1.

Accepted Answer

x = 0, x = 2

Answer

x = 1, x = 3

Answer

x = -1, x = 1

Question 52

Find the absolute maximum of f(x) = -x^2 + 4x + 5 on the interval [-1, 3].

Accepted Answer

9

Answer

5

Answer

13

Answer

1

Question 53

Find the derivative of f(x) = sin(2x).

Accepted Answer

2cos(2x)

Answer

-sin(2x)

Answer

-2cos(2x)

Answer

sin(2x)

Question 54

Find the equation of the normal line to the graph of f(x) = x^3 + 2x - 1 at the point (1, 2).

Accepted Answer

y = -x + 3

Answer

y = -x + 1

Answer

y = x - 1

Answer

y = x + 3

Question 55

Find the rate of change of the function f(x) = 3x^2 - 2x + 1 with respect to x.

Accepted Answer

6x - 2

Answer

2x - 1

Answer

3x

Question 56

Find the derivative of the function f(x) = e^(2x).

Accepted Answer

2e^(2x)

Answer

e^(2x)

Answer

e^(x^2)

Answer

2x*e^(2x)

Question 57

Find the derivative of the function f(x) = ln(x^2 + 1).

Accepted Answer

2x/(x^2 + 1)

Answer

1/(x^2 + 1)

Answer

2/(x^2 + 1)

Answer

(2x)/(x^2 - 1)

Question 58

Find the second derivative of the function f(x) = x^4 - 2x^2 + 3.

Accepted Answer

12x - 4

Answer

x^4 - 2

Answer

x^3 - 2

Answer

4x^3 - 4x