AP Calculus BC Practice Quiz: The Product Rule

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

Test your understanding with short quizzes. This quiz has 7 questions to check your progress.

Question 1 of 7

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

A)6x^2 - 6x + 2B)4xC)2x^2 - 6x - 2D)6x^2 + 6x - 2

All Questions (7)

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

A) 6x^2 - 6x + 2

B) 4x

C) 2x^2 - 6x - 2

D) 6x^2 + 6x - 2

Correct Answer: A

To find the derivative of the product of two functions, u(x) = x^2 + 1 and v(x) = 2x - 3, we use the product rule: f'(x) = u'(x)v(x) + u(x)v'(x). Here, u'(x) = 2x and v'(x) = 2. Applying the rule: f'(x) = (2x)(2x - 3) + (x^2 + 1)(2) = 4x^2 - 6x + 2x^2 + 2 = 6x^2 - 6x + 2.

If y = x^3 * cos(x), what is dy/dx?

A) -3x^2 * sin(x)

B) 3x^2 * cos(x) - x^3 * sin(x)

C) 3x^2 * cos(x) + x^3 * sin(x)

D) 3x^2 * sin(x) - x^3 * cos(x)

Correct Answer: B

Using the product rule, d/dx[f(x)g(x)] = f'(x)g(x) + f(x)g'(x). Let f(x) = x^3 and g(x) = cos(x). Then f'(x) = 3x^2 and g'(x) = -sin(x). Applying the formula: dy/dx = (3x^2)(cos(x)) + (x^3)(-sin(x)) = 3x^2 * cos(x) - x^3 * sin(x).

Let h(x) = f(x)g(x), where f and g are differentiable functions. The table below gives values of f, g, and their derivatives at x = 2. x | f(x) | f'(x) | g(x) | g'(x) --|---|---|---|--- 2 | 4 | -1 | 5 | 3 What is the value of h'(2)?

A) -3

B) 2

C) 7

D) 20

Correct Answer: C

The product rule states that h'(x) = f'(x)g(x) + f(x)g'(x). To find h'(2), we substitute the values from the table at x = 2: h'(2) = f'(2)g(2) + f(2)g'(2) = (-1)(5) + (4)(3) = -5 + 12 = 7.

Find the derivative of y = e^x * sin(x).

A) e^x * cos(x)

B) e^x (sin(x) - cos(x))

C) e^x (sin(x) + cos(x))

D) e^x * sin(x) + cos(x)

Correct Answer: C

Apply the product rule with f(x) = e^x and g(x) = sin(x). We have f'(x) = e^x and g'(x) = cos(x). The derivative is f'(x)g(x) + f(x)g'(x) = (e^x)(sin(x)) + (e^x)(cos(x)). Factoring out e^x gives e^x(sin(x) + cos(x)).

What is the slope of the line tangent to the curve f(x) = (x^2 + 3x) * ln(x) at x = 1?

A) 0

B) 4

C) 5

D) e

Correct Answer: B

The slope of the tangent line is the value of the derivative at that point. First, find f'(x) using the product rule. Let u(x) = x^2 + 3x and v(x) = ln(x). Then u'(x) = 2x + 3 and v'(x) = 1/x. So, f'(x) = (2x + 3)(ln(x)) + (x^2 + 3x)(1/x). Now, evaluate at x = 1: f'(1) = (2(1) + 3)(ln(1)) + (1^2 + 3(1))(1/1) = (5)(0) + (4)(1) = 0 + 4 = 4.

If f(x) and g(x) are differentiable functions, which of the following is the derivative of the product h(x) = f(x)g(x)?

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

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

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

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

Correct Answer: B

This question asks for the definition of the product rule. The derivative of a product of two differentiable functions f(x) and g(x) is given by the formula f'(x)g(x) + f(x)g'(x). Option A is a common mistake, while options C and D are variations of the quotient rule.

Let f(x) = (ax^2+b)(cx+d), where a, b, c, and d are constants. Which of the following is f'(x)?

A) (2ax)(c)

B) (2ax)(cx+d) + c(ax^2+b)

C) (2ax+b)(c)

D) 2acx

Correct Answer: B

This is a direct application of the product rule with general constants. Let u(x) = ax^2+b and v(x) = cx+d. The derivatives are u'(x) = 2ax and v'(x) = c. According to the product rule, f'(x) = u'(x)v(x) + u(x)v'(x), which is (2ax)(cx+d) + (ax^2+b)(c). Option A represents the common error of multiplying the derivatives (u'v').