AP Calculus BC 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

Let f(x) = 8x³. Which of the following is f'(x)?

A)24x²B)8x²C)24x³D)3x²

All Questions (9)

Let f(x) = 8x³. Which of the following is f'(x)?

A) 24x²

B) 8x²

C) 24x³

D) 3x²

Correct Answer: A

To find the derivative of f(x) = 8x³, we use the constant multiple rule and the power rule. The constant multiple rule allows us to carry the 8, and the power rule states that the derivative of x³ is 3x². Therefore, f'(x) = 8 * (3x²) = 24x².

Find the derivative of the function y = x⁴ - 10.

A) 4x³ - 10

B) 4x³

C) x³ - 10

D) 4x⁴

Correct Answer: B

Using the difference rule, we can differentiate each term separately. The derivative of x⁴ is 4x³ by the power rule. The derivative of a constant, -10, is 0. Therefore, dy/dx = 4x³ - 0 = 4x³.

If f(x) = 2x³ - 7x + 5, what is f'(x)?

A) 6x² - 7

B) 3x² - 7

C) 6x³ - 7x

D) 6x² - 7x + 5

Correct Answer: A

This problem requires using the sum, difference, constant multiple, and power rules on a polynomial function. Differentiating term by term: the derivative of 2x³ is 6x², the derivative of -7x is -7, and the derivative of the constant 5 is 0. Combining these results gives f'(x) = 6x² - 7.

The process of finding the derivative of g(x) = f(x) + k(x) by calculating f'(x) + k'(x) is an application of which rule?

A) The Power Rule

B) The Constant Multiple Rule

C) The Sum Rule

D) The Difference Rule

Correct Answer: C

The Sum Rule states that the derivative of a sum of functions is the sum of their derivatives. This is precisely the procedure described: d/dx [f(x) + k(x)] = f'(x) + k'(x).

Given the polynomial function g(t) = 4t³ - 9t² + 2t, what is the value of g'(2)?

A) 14

B) 16

C) 26

D) 48

Correct Answer: A

First, find the derivative of g(t) using the sum, difference, and constant multiple rules combined with the power rule: g'(t) = 12t² - 18t + 2. Next, substitute t = 2 into the derivative: g'(2) = 12(2)² - 18(2) + 2 = 12(4) - 36 + 2 = 48 - 36 + 2 = 14.

Let f and g be differentiable functions. If h(x) = 5f(x) - 2g(x), and it is known that f'(-1) = 4 and g'(-1) = 3, what is the value of h'(-1)?

A) 11

B) 14

C) 26

D) 1

Correct Answer: B

Using the difference and constant multiple rules, we find the derivative of h(x) is h'(x) = 5f'(x) - 2g'(x). To find h'(-1), we substitute the given values: h'(-1) = 5f'(-1) - 2g'(-1) = 5(4) - 2(3) = 20 - 6 = 14.

Let f(x) = ax⁴ - 6x², where a is a constant. If f'(1) = 8, what is the value of a?

A) 1

B) 2

C) 4

D) 5

Correct Answer: D

First, find the derivative of f(x) with respect to x: f'(x) = 4ax³ - 12x. Now, use the given condition that f'(1) = 8. Substitute x = 1 into the derivative: f'(1) = 4a(1)³ - 12(1) = 4a - 12. Set this equal to 8: 4a - 12 = 8. Solving for a, we get 4a = 20, so a = 5.

Which of the following expressions is equivalent to d/dx [c⋅f(x) - g(x)], where c is a constant?

A) c⋅f'(x) - g'(x)

B) f'(x) - g'(x)

C) c⋅f'(x) + g'(x)

D) c⋅f(x) - g'(x)

Correct Answer: A

This requires applying both the constant multiple rule and the difference rule. The derivative of c⋅f(x) is c⋅f'(x) by the constant multiple rule. The derivative of g(x) is g'(x). The difference rule states that we subtract the derivatives. Therefore, d/dx [c⋅f(x) - g(x)] = c⋅f'(x) - g'(x).

The derivative of a polynomial function p(x) is p'(x) = 15x² + 4. Which of the following could be p(x)?

A) 3x³ + 4x - 1

B) 5x³ + 4x + 7

C) 15x³ + 4x

D) 30x + 4

Correct Answer: B

We need to find a function whose derivative is 15x² + 4. We can check each option. A) d/dx(3x³ + 4x - 1) = 9x² + 4. B) d/dx(5x³ + 4x + 7) = 15x² + 4. C) d/dx(15x³ + 4x) = 45x² + 4. D) d/dx(30x + 4) = 30. Option B is the correct choice, as its derivative matches the given p'(x). The constant term (+7) differentiates to zero, which is consistent with the process.