Differentiation in Calculus: Master Derivatives with Practice Quizzes

Question 1

Which of the following is NOT an application of differentiation?

Accepted Answer

Finding the volume of a sphere

Answer

Finding the velocity of an object

Answer

Finding the area under a curve

Answer

Finding the critical points of a function

Question 2

The derivative of a constant function is:

Accepted Answer

0

Answer

1

Answer

The constant itself

Answer

Undefined

Question 3

What is the product rule for differentiation?

Accepted Answer

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

Answer

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

Question 4

Which statement about differentiation is correct?

Accepted Answer

Differentiation can be used to find the slope of a tangent line to a curve.

Answer

Differentiation is the process of finding the integral of a function.

Answer

Differentiation is a linear operation.

Answer

Differentiation is only applicable to polynomial functions.

Question 5

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

Accepted Answer

3x^2 + 4x

Answer

x^2 + 4x - 5

Answer

9x^2 + 4x

Answer

3x^2 - 5

Question 6

Which of the following is NOT a technique for finding derivatives?

Accepted Answer

Integral Rule

Answer

Power Rule

Answer

Product Rule

Answer

Quotient Rule

Question 7

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

Accepted Answer

2e^(2x-1)

Answer

e^(2x-1)

Answer

2x e^(2x-1)

Answer

-e^(2x-1)

Question 8

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

Accepted Answer

2x/(x^2 + 1)

Answer

1/(x^2 + 1)

Answer

x^2/(x^2 + 1)

Answer

(2x + 1)/(x^2 + 1)

Question 9

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

Accepted Answer

1/(1+x^2)

Answer

-1/(1+x^2)

Answer

(1+x^2)

Answer

1

Question 10

To find the derivative of h(x) = sin(x^2 + 1) using the chain rule, what should be the first step?

Accepted Answer

Find the derivative of x^2 + 1.

Answer

Find the derivative of sin(x).

Answer

Substitute x^2 + 1 for sin(x).

Answer

Apply the power rule to x^2 + 1.

Question 11

Which of the following is not a technique for finding derivatives?

Accepted Answer

Integration by Parts

Answer

Power Rule

Answer

Product Rule

Answer

Chain Rule

Question 12

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

Accepted Answer

3x^2

Answer

x^2

Answer

3x

Answer

x^4

Question 13

Which of the following accurately represents the product rule for differentiation, assuming f(x) = g(x) * h(x)?

Accepted Answer

f'(x) = g(x) * h'(x) + g'(x) * h(x)

Answer

f'(x) = g'(x) * h'(x)

Answer

f'(x) = g'(x) * h(x) - g(x) * h'(x)

Answer

f'(x) = g(x) * h(x)

Question 14

Utilize the chain rule to find the derivative of the function f(x) = e^(2x).

Accepted Answer

e^(2x) * 2x

Answer

2e^(2x)

Answer

e^(2x) / 2

Answer

2e^(x)

Question 15

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

Accepted Answer

2

Answer

2x

Answer

x^2

Answer

0

Question 16

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

Accepted Answer

3x^2 + 4x - 5

Answer

x^2 + 4x - 5

Answer

3x^2 - 5x + 1

Answer

x^2 - 5x + 1

Question 17

Which of the following correctly defines the purpose of the chain rule in differentiation?

Accepted Answer

Differentiating a function composed of other functions

Answer

Finding the critical points of a function

Answer

Integrating a function

Answer

Solving differential equations

Question 18

Utilize the chain rule to determine the derivative of the function f(x) = sin(x^2).

Accepted Answer

2x*cos(x^2)

Answer

cos(x^2)

Answer

x*cos(x^2)

Answer

sin(2x)

Question 19

Calculate the derivative of the function f(x) = |x|.

Accepted Answer

Not differentiable at x = 0

Answer

1

Answer

-1

Answer

x/|x|

Question 20

Which method is NOT utilized for finding derivatives?

Accepted Answer

Integration by Parts

Answer

Power Rule

Answer

Quotient Rule

Answer

Chain Rule

Question 21

Find the derivative of the function f(x) = ln(x) using appropriate rules of differentiation.

Accepted Answer

1/x

Answer

x

Answer

ln(x)

Answer

e^x

Question 22

Apply the chain rule to calculate the derivative of the function f(x) = e^(x^2 + 1).

Accepted Answer

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

Answer

x*e^(x^2 + 1)

Answer

e^(x^2 + 1)

Answer

Not defined

Question 23

Find the derivative of the function f(x) = sin(x²).

Accepted Answer

cos(x²)*2x

Answer

sin(x²)

Answer

cos(x²)*x

Answer

-cos(x²)*2x

Question 24

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

Accepted Answer

e^x

Answer

x^e

Answer

e

Answer

0

Question 25

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

Accepted Answer

1/x

Answer

x

Answer

ln(x)

Answer

0

Question 26

Which of the following is the second derivative of the function f(x) = x^4 - 2x² + 1?

Accepted Answer

12x² - 4

Answer

4x³ - 4x

Answer

24x - 4

Answer

16x² - 4

Question 27

Find the derivative of the function f(x) = x³ + 2x² - 5x + 1 using the power rule. Make sure to simplify your answer.

Accepted Answer

3x² + 4x - 5

Answer

x² + 4x - 5

Answer

3x² - 4x - 5

Answer

x² - 4x - 5

Question 28

Find the derivative of the function f(x) = (x² + 1)(x - 2) using the product rule. Express your answer in the simplest form.

Accepted Answer

2x(x - 2) + (x² + 1)

Answer

(x² + 1)(x - 2) + (x² + 1)

Answer

2x(x - 2) - (x² + 1)

Answer

2x(x - 2)

Question 29

Find the derivative of the function f(x) = sin(x²) using the chain rule. Write your answer in its simplest form.

Accepted Answer

cos(x²) * 2x

Answer

sin(x²) * 2x

Answer

cos(x²)

Answer

sin(x²)

Question 30

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

Accepted Answer

f'(x) = 3x^2 + 4x - 5

Answer

f'(x) = x^2 + 4x - 5

Answer

f'(x) = 3x^3 + 4x - 5

Answer

f'(x) = 3x^2 + 2x - 5

Question 31

Find the derivative of f(x) = sin(2x + 1) using the chain rule.

Accepted Answer

f'(x) = 2cos(2x + 1)

Answer

f'(x) = 2sin(2x + 1)

Answer

f'(x) = cos(2x + 1)

Answer

f'(x) = sin(2x + 1)

Question 32

Determine if the following statement is true or false: 'The derivative of a constant function is always zero.'

Accepted Answer

True

Answer

Sometimes true, sometimes false

Answer

Depends on the constant

Answer

False

Question 33

Consider the function f(x) = ln(x^2 + 1). Which of the following represents the correct derivative of f(x)?

Accepted Answer

f'(x) = 2x/(x^2 + 1)

Answer

f'(x) = (x^2 + 1)/x

Answer

f'(x) = x/(x^2 + 1)

Question 34

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

Accepted Answer

2cos(2x)

Answer

2sin(2x)

Answer

cos(2x)

Answer

-2sin(2x)

Question 35

Which of the following is the local maximum of the function f(x) = -x^2 + 4x - 3?

Accepted Answer

x = 2

Answer

x = 0

Answer

x = -2

Question 36

The derivative of the inverse function of f(x) = x^3 is:

Accepted Answer

1/3x^(2/3)

Answer

x^(2/3)

Answer

3x^2

Answer

-3x^(2/3)

Question 37

Which of the following statements about differentiation is true?

Accepted Answer

It is a process of finding the instantaneous rate of change of a function.

Answer

It is only used in abstract mathematics.

Answer

It is not applicable to real-world problems.

Question 38

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

Accepted Answer

f'(x) = 2cos(2x)

Answer

f'(x) = -2cos(2x)

Answer

f'(x) = sin(x)

Answer

f'(x) = 2sin(x)

Question 39

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

Accepted Answer

f'(x) = e^x

Answer

f'(x) = xe^x

Answer

f'(x) = 2e^x

Answer

f'(x) = x^2e^x

Question 40

Calculate the rate of change of y with respect to x for the function f(x, y) = xy + 2x - y^2 at the point (1, 2).

Accepted Answer

Rate of change = 4

Answer

Rate of change = 1

Answer

No information is provided in the question to calculate the rate of change.

Question 41

Find the derivative of the function f(x) = 2x³ - 5x² + 7x - 1 with respect to x.

Accepted Answer

6x² - 10x + 7

Answer

2x³ + 5x² + 7x - 1

Answer

2x² - 5 + 7

Answer

6x³ - 5x² - 7x - 1

Question 42

Find the derivative of the function h(x) = sin(3x) using the chain rule.

Accepted Answer

3cos(3x)

Answer

cos(3x)

Answer

sin(x)

Answer

-3cos(3x)

Question 43

Which of the following is the derivative of the function f(x) = e^(2x - 1)?

Accepted Answer

2e^(2x - 1)

Answer

2xe^(2x - 1)

Answer

e^(2x + 1)

Answer

e^(2x)

Question 44

Which of the following methods can be used to find the derivative of an implicit function?

Accepted Answer

Implicit differentiation

Answer

Product rule

Answer

Power rule

Answer

Quotient rule

Question 45

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

Accepted Answer

2e^(2x + 1)

Answer

-2e^(2x + 1)

Answer

e^(2x + 1)

Answer

2x * e^(2x + 1)

Question 46

Find the derivative of the function *f(x) = arctan(x)*.

Accepted Answer

1 / (1 + x²)

Answer

-x / (1 + x²)

Answer

-1 / (1 + x²)

Answer

x / (1 + x²)

Question 47

Which of the following is the derivative of the function f(x) = x^3 + 2x^2 + 1?

Accepted Answer

3x^2 + 4x

Answer

3x^2 - 4x

Answer

x^2 + 4x

Answer

x^2 - 4x

Question 48

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

Accepted Answer

2x - 1

Answer

x^2 - 3x + 2

Answer

x^2 + 3x - 2

Answer

x^2 - 3x - 2

Question 49

Calculate the derivative of the function h(x) = sin(2x).

Accepted Answer

2cos(2x)

Answer

-2cos(2x)

Answer

sin(2x)

Answer

cos(2x)

Question 50

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

Accepted Answer

x = 0, x = 2

Answer

x = -1, x = 2

Answer

x = 0, x = 1

Answer

x = 1, x = 3

Question 51

Determine the extrema (maximum and minimum) of the function g(x) = -x^2 + 4x - 3.

Accepted Answer

Maximum at (2, 1), Minimum at (0, -3)

Answer

Maximum at (1, 2), Minimum at (3, 0)

Answer

Maximum at (3, 0), Minimum at (1, 2)

Answer

Maximum at (0, -3), Minimum at (2, 1)

Question 52

Which of the following is the equation of the tangent line to the curve y = x^2 - 2x + 1 at the point (1, 0)?

Accepted Answer

y = -2x + 3

Answer

y = x

Answer

y = x^2 - 2x

Answer

y = 2x - 1

Question 53

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

Accepted Answer

1 for x > 0, -1 for x < 0

Answer

1 for x ≥ 0, -1 for x < 0

Answer

Undefined at x = 0, 1 for x > 0, -1 for x < 0

Answer

0 for x = 0, 1 for x > 0, -1 for x < 0

Question 54

Calculate the second derivative of the function g(x) = x^3 + 2x^2.

Accepted Answer

6x + 4

Answer

x^3 + 4x

Answer

x^4 + 4x^2

Answer

3x^2 + 4x

Question 55

Which of the following is an application of differentiation in real-world scenarios?

Accepted Answer

Finding the rate of change of a quantity with respect to another quantity

Answer

Solving polynomial equations

Answer

Finding the area under a curve

Question 56

Determine the concavity of the function f(x) = x^4 - 2x^2 + 1.

Accepted Answer

Concave up for x > 0, Concave down for x < 0

Answer

Concave up for all x

Answer

Concave down for all x

Answer

Concave down for x > 0, Concave up for x < 0