AP Calculus AB Practice Quiz: Derivative Rules: Constant, Sum, Difference, and Constant Multiple

Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026

Test your understanding with short quizzes. This quiz has 9 questions to check your progress.

Question 1 of 9

What is the derivative of the function f(x) = 7x^4?

A)28x^3B)7x^3C)28x^4D)4x^3

All Questions (9)

What is the derivative of the function f(x) = 7x^4?

A) 28x^3

B) 7x^3

C) 28x^4

D) 4x^3

Correct Answer: A

Using the constant multiple rule and the power rule, the derivative of x^4 is 4x^3. This is then multiplied by the constant 7, resulting in 7 * 4x^3 = 28x^3.

Find the derivative of g(x) = x^5 + 3x^2.

A) 5x^4 + 3x

B) 5x^4 + 6x

C) 4x^4 + 6x

D) x^4 + 6x

Correct Answer: B

Using the sum rule, each term is differentiated separately. The derivative of x^5 is 5x^4 by the power rule. The derivative of 3x^2 is 3 * 2x = 6x by the constant multiple and power rules. The sum of these derivatives is 5x^4 + 6x.

If y = 4x^3 - 5x^2 + 2, what is dy/dx?

A) 12x^2 - 10x + 2

B) 12x^2 - 10x

C) 7x^2 - 7x

D) 4x^2 - 5x

Correct Answer: B

The derivative of the polynomial is found by applying the sum and difference rules to each term. The derivative of 4x^3 is 12x^2. The derivative of -5x^2 is -10x. The derivative of the constant 2 is 0. Combining these results gives 12x^2 - 10x.

Let f(x) be a differentiable function. What is the derivative of g(x) = 5f(x) - 10?

A) 5f'(x) - 10

B) 5f(x)

C) 5f'(x)

D) f'(x) - 10

Correct Answer: C

This question tests the combination of the constant multiple, difference, and constant rules. Using the difference rule, we differentiate 5f(x) and 10 separately. The derivative of 5f(x) is 5f'(x) by the constant multiple rule. The derivative of the constant 10 is 0. Therefore, g'(x) = 5f'(x) - 0 = 5f'(x).

Let f(x) = 2x^3 - 4x + 1. What is the value of f'(2)?

A) 16

B) 20

C) 9

D) 24

Correct Answer: B

First, find the derivative function, f'(x), using the power, constant multiple, and sum/difference rules. f'(x) = d/dx(2x^3 - 4x + 1) = 6x^2 - 4. Then, substitute x = 2 into the derivative: f'(2) = 6(2)^2 - 4 = 6(4) - 4 = 24 - 4 = 20.

Let h(x) = 2f(x) + 3g(x). If f'(5) = -4 and g'(5) = 2, what is the value of h'(5)?

A) -2

B) 2

C) -14

D) -8

Correct Answer: A

According to the sum and constant multiple rules, the derivative of h(x) is h'(x) = 2f'(x) + 3g'(x). To find h'(5), substitute the given values: h'(5) = 2f'(5) + 3g'(5) = 2(-4) + 3(2) = -8 + 6 = -2.

The derivative of the function f(x) = kx^2 - 3x is f'(x) = 10x - 3. What is the value of the constant k?

A) 10

B) 2

C) 5

D) 20

Correct Answer: C

First, find the derivative of f(x) = kx^2 - 3x using the derivative rules. The derivative of kx^2 is 2kx (constant multiple and power rules). The derivative of -3x is -3. So, f'(x) = 2kx - 3. We are given that f'(x) = 10x - 3. By comparing the two expressions, we can see that 2k must be equal to 10. Solving for k: 2k = 10, so k = 5.

Which of the following represents the derivative of the polynomial function p(x) = -x^4 + 2x^3 - 6x + 7?

A) -4x^3 + 6x^2 - 6

B) -4x^3 + 6x^2 - 6x + 7

C) -x^3 + 2x^2 - 6

D) -4x^5 + 6x^4 - 6x^2

Correct Answer: A

Differentiate the polynomial term by term using the power, constant multiple, and sum/difference rules. The derivative of -x^4 is -4x^3. The derivative of 2x^3 is 6x^2. The derivative of -6x is -6. The derivative of the constant 7 is 0. Combining these results gives p'(x) = -4x^3 + 6x^2 - 6.

For the function f(x) = (1/3)x^3 - 2x^2 + 3x - 1, find all values of x for which f'(x) = 0.

A) x = 1 and x = 3

B) x = -1 and x = -3

C) x = 0 and x = 2

D) x = 3 only

Correct Answer: A

First, find the derivative of f(x) using the power and sum/difference rules: f'(x) = (1/3)(3x^2) - 2(2x) + 3(1) - 0, which simplifies to f'(x) = x^2 - 4x + 3. Next, set the derivative equal to zero: x^2 - 4x + 3 = 0. This is a quadratic equation that can be factored as (x - 1)(x - 3) = 0. The solutions are x = 1 and x = 3.