Master Differentiation in Calculus: Interactive Quizzes and Practice

Question 1

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

Accepted Answer

2x + 2

Answer

x^2 + 2

Answer

2x - 3

Answer

x - 3

Question 2

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

Accepted Answer

cos(x)

Answer

-sin(x)

Answer

tan(x)

Answer

sec(x)

Question 3

Which method can be employed to calculate the area under the curve of a function?

Accepted Answer

Integration

Answer

Differentiation

Answer

Trigonometry

Answer

Algebra

Question 4

Find the integral of the function f(x) = 2x + 1.

Accepted Answer

x^2 + x + C

Answer

2x^2 + C

Answer

2x + C

Answer

x + C

Question 5

Which of the following is not a differentiation rule for trigonometric functions?

Accepted Answer

d/dx(cot x) = -sin^2 x

Answer

d/dx(sin x) = cos x

Answer

d/dx(cos x) = -sin x

Answer

d/dx(tan x) = sec^2 x

Question 6

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

Accepted Answer

1/x

Answer

x

Answer

ln 10 * x

Answer

-1/x

Question 7

Utilizing the quotient rule, find the derivative of the function f(x) = (x^2 + 1)/(x - 1).

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 8

Which of the following demonstrates a real-world application of derivatives?

Accepted Answer

All of the provided options

Answer

Determining the rate of change of a specific population

Answer

Calculating the slope of a projectile's trajectory

Answer

Maximizing profits within a business scenario

Question 9

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

Accepted Answer

2x + 3

Answer

x^2 + 3

Answer

2x

Answer

x + 3

Question 10

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

Accepted Answer

cos(x)

Answer

-sin(x)

Answer

sin(x)

Answer

cos(x)/sin(x)

Question 11

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

Accepted Answer

3x^2

Answer

6x

Answer

2x^2

Answer

x^3

Question 12

Find the derivative of f(x) = x³ + 2x² - 5x + 1, where x is a real number.

Accepted Answer

3x² + 4x - 5

Answer

x² + 4x - 5

Answer

3x² - 4x - 5

Answer

x² - 4x - 5

Question 13

Find the derivative of f(x) = cos(x), where x is a real number.

Accepted Answer

-sin(x)

Answer

sin(x)

Answer

-cos(x)

Answer

cos(x)

Question 14

Find the equation of the tangent to the curve y = x² - 3x + 2 at the point (1, 0).

Accepted Answer

y = -x + 1

Answer

y = x + 1

Answer

y = -2x + 1

Answer

y = 2x + 1

Question 15

Which of the following functions has a constant rate of change?

Accepted Answer

f(x) = 2x + 1

Answer

f(x) = x²

Answer

f(x) = x³

Answer

f(x) = e^x

Question 16

Calculate the derivative of the function y = (x^2 + 3x)/(x - 1).

Accepted Answer

5x - 1

Answer

5x - 3

Answer

5x + 3

Answer

5x + 1

Question 17

Determine the derivative of f(x) = x^2.

Accepted Answer

2x

Answer

x

Answer

x^3

Answer

2x^2

Question 18

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

Accepted Answer

cos(x)

Answer

-sin(x)

Answer

tan(x)

Answer

cot(x)

Question 19

Identify the derivative of f(x) = e^x.

Accepted Answer

e^x

Answer

xe^x

Answer

e^(x-1)

Answer

1/e^x

Question 20

Which application of differentiation is used to determine the velocity of an object?

Accepted Answer

Finding the rate of change

Answer

Locating critical points

Answer

Sketching graphs

Answer

Solving differential equations

Question 21

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

Accepted Answer

-2cos(x)

Answer

2cos(x)

Answer

sin(x)

Answer

-sin(x)

Question 22

To find the derivative of f(x) = x^3 - 2x^2 + x - 1, use the power rule. What is the result?

Accepted Answer

3x^2 - 4x + 1

Answer

x^2 - 4x + 1

Answer

3x^2 - 2x + 1

Answer

x^2 - 2x + 1

Question 23

The product rule for differentiation states that when differentiating f(x)g(x), you should:

Accepted Answer

Multiply the derivatives of f(x) and g(x)

Answer

Add the derivatives of f(x) and g(x)

Answer

Subtract the derivatives of f(x) and g(x)

Answer

Divide the derivative of f(x) by the derivative of g(x)

Question 24

Which of the following is NOT an application of differentiation?

Accepted Answer

Solving algebraic equations

Answer

Optimization problems

Answer

Motion analysis

Answer

Curve fitting

Question 25

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

Accepted Answer

3e^(3x)

Answer

e^(3x)

Answer

1/3e^(3x)

Answer

-3e^(3x)

Question 26

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

Accepted Answer

(3x^2 + 2)/(2sqrt(x^3 + 2x))

Answer

(3x^2 - 2)/(2sqrt(x^3 + 2x))

Answer

-(3x^2 + 2)/(2sqrt(x^3 + 2x))

Answer

-(3x^2 - 2)/(2sqrt(x^3 + 2x))

Question 27

Find the derivative of y = ln(sin(x) + cos(x)).

Accepted Answer

(cos(x) - sin(x))/(sin(x) + cos(x))

Answer

(sin(x) + cos(x))/(sin(x) - cos(x))

Answer

-(cos(x) - sin(x))/(sin(x) + cos(x))

Answer

-(sin(x) + cos(x))/(sin(x) - cos(x))

Question 28

For L'Hopital's rule to be applicable in evaluating a limit, which condition must be met?

Accepted Answer

The limit of the expression is 0/0 or ∞/∞.

Answer

The expression is continuous at the point of evaluation.

Answer

The expression has a local minimum or maximum at the point of evaluation.

Answer

The derivative of the denominator of the expression is zero at the point of evaluation.

Question 29

Find the rate of change of the volume of a sphere with respect to its radius, if the volume of the sphere is given by V = (4/3)πr^3.

Accepted Answer

4πr^2

Answer

2πr

Answer

πr^3

Answer

πr^2

Question 30

Which of the following is the derivative of the logarithmic function y = log₂(x)?

Accepted Answer

1/(x*ln(2))

Answer

2^(x-1)

Answer

1/x

Answer

ln(2)/x

Question 31

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

Accepted Answer

f'(x) = e^x

Answer

f'(x) = xe^x

Answer

f'(x) = -e^x

Question 32

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

Accepted Answer

f'(x) = 1/x

Answer

f'(x) = ln(x+1)

Answer

f'(x) = x

Answer

f'(x) = -1/x

Question 33

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

Accepted Answer

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

Answer

f'(x) = 1/(x - 1)

Answer

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

Answer

f'(x) = 2x/(x - 1)

Question 34

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

Accepted Answer

f'(x) = 2xe^(x^2 - 1)

Answer

f'(x) = x^2e^(x^2 - 1)

Answer

f'(x) = e^(x^2 - 1)

Answer

f'(x) = e^(2x - 1)

Question 35

Which of the following accurately defines the derivative of a function?

Accepted Answer

The instantaneous rate of change of a function with respect to its independent variable

Answer

The area bounded by the graph of a function and the x-axis

Answer

The maximum value that a function can attain

Question 36

Apply the chain rule to determine the derivative of f(x) = sin(x²).

Accepted Answer

2x * cos(x²)

Answer

sin(2x)

Answer

2cos(x²)

Answer

cos(x²)

Question 37

Which technique is most appropriate for finding the derivative of an implicit function like y² + x² = 25?

Accepted Answer

Implicit differentiation

Answer

Numerical differentiation

Answer

Quotient rule

Answer

Logarithmic differentiation

Question 38

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

Accepted Answer

(2x + cos(x)) * e^(x² + sin(x))

Answer

sin(x) * e^(x² + sin(x))

Answer

x * e^(x² + sin(x))

Answer

cos(x) * e^(x² + sin(x))

Question 39

Find the derivative of f(x) = x³ - 2x + 5

Accepted Answer

f'(x) = 3x² - 2

Answer

f'(x) = 3x

Answer

f'(x) = 5 - 2x

Answer

f'(x) = x³ - 2

Question 40

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

Accepted Answer

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

Answer

f'(x) = cos(x)

Answer

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

Answer

f'(x) = sin(x) + cos(x)

Question 41

What is the derivative of f(x) = e^(2x)?

Accepted Answer

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

Answer

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

Answer

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

Question 42

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

Accepted Answer

f'(x) = 1/x

Answer

f'(x) = ln(x)

Answer

f'(x) = x

Answer

f'(x) = 0

Question 43

Which of the following statements correctly defines the Chain Rule for differentiation?

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 44

Determine the derivative of the function f(x) = (x² + 1)³.

Accepted Answer

f'(x) = 3(x² + 1)² * 2x

Answer

f'(x) = (x² + 1)²

Answer

f'(x) = 3x²(x² + 1)

Answer

f'(x) = 3(x² + 1) * 2x

Question 45

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

Accepted Answer

f'(x) = 1 for x > 0, -1 for x < 0

Answer

f'(x) = x

Answer

f'(x) = |1|

Answer

f'(x) = 0

Question 46

Find the derivative of the square root function f(x) = √(x).

Accepted Answer

f'(x) = 1 / (2√(x))

Answer

f'(x) = √(x)

Answer

f'(x) = 0

Answer

f'(x) = x√(x)

Question 47

What is the derivative of the function  f(x) = 3x² + 2x - 1?

Accepted Answer

6x + 2

Answer

9x + 2

Answer

6x² + 2x

Answer

3x + 2

Question 48

What is the gradient of the tangent line to the curve y = x² - 3x + 2 at the point (1, 0)?

Accepted Answer

-1

Answer

3

Answer

-3

Answer

1

Question 49

What is the derivative of h(x) = (2x + 1)³?

Accepted Answer

6(2x + 1)²

Answer

3(2x + 1)³

Answer

6(2x + 1)³

Answer

3(2x + 1)²

Question 50

What are the critical points of the function f(x) = x³ - 6x² + 9x?

Accepted Answer

x = 1, x = 3

Answer

x = 0, x = 3

Answer

x = 1, x = 2

Answer

x = 0, x = 2

Question 51

Which of the following functions has a derivative that is always positive?

Accepted Answer

f(x) = e^x

Answer

f(x) = x²

Answer

f(x) = sin(x)

Answer

f(x) = cos(x)