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AP Calculus BC Practice Quiz: The Quotient 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) = (2x + 3) / (x - 1). Find f'(x).

All Questions (7)

Let f(x) = (2x + 3) / (x - 1). Find f'(x).

A) f'(x) = -5 / (x - 1)^2

B) f'(x) = 2 / 1

C) f'(x) = (4x + 1) / (x - 1)^2

D) f'(x) = -5 / (x - 1)

Correct Answer: A

To find the derivative of f(x), we use the quotient rule: d/dx [g(x)/h(x)] = [h(x)g'(x) - g(x)h'(x)] / [h(x)]^2. Here, g(x) = 2x + 3 and h(x) = x - 1. Their derivatives are g'(x) = 2 and h'(x) = 1. Plugging into the formula: f'(x) = [(x - 1)(2) - (2x + 3)(1)] / (x - 1)^2 = [2x - 2 - 2x - 3] / (x - 1)^2 = -5 / (x - 1)^2.

Let y = (x^2) / (3x - 5). What is dy/dx?

A) (9x^2 - 10x) / (3x - 5)^2

B) (3x^2 - 10x) / (3x - 5)^2

C) 2x / 3

D) (-3x^2 + 10x) / (3x - 5)^2

Correct Answer: B

Using the quotient rule, d/dx [g(x)/h(x)] = [h(x)g'(x) - g(x)h'(x)] / [h(x)]^2, with g(x) = x^2 and h(x) = 3x - 5. We find g'(x) = 2x and h'(x) = 3. Applying the rule: dy/dx = [(3x - 5)(2x) - (x^2)(3)] / (3x - 5)^2 = [6x^2 - 10x - 3x^2] / (3x - 5)^2 = (3x^2 - 10x) / (3x - 5)^2.

If h(x) = (x + 4) / (x^2 + 1), what is the value of h'(1)?

A) -1/2

B) -2

C) 1/4

D) 0

Correct Answer: B

First, find the derivative h'(x) using the quotient rule. Let g(x) = x + 4 and f(x) = x^2 + 1. Then g'(x) = 1 and f'(x) = 2x. So, h'(x) = [(x^2 + 1)(1) - (x + 4)(2x)] / (x^2 + 1)^2 = [x^2 + 1 - 2x^2 - 8x] / (x^2 + 1)^2 = (-x^2 - 8x + 1) / (x^2 + 1)^2. Now, evaluate at x = 1: h'(1) = (-(1)^2 - 8(1) + 1) / ((1)^2 + 1)^2 = (-1 - 8 + 1) / (2)^2 = -8 / 4 = -2.

Let f and g be differentiable functions and let H(x) = f(x) / g(x). If f(2) = 3, f'(2) = -1, g(2) = 5, and g'(2) = 4, what is the value of H'(2)?

A) -17/25

B) 7/25

C) -1/4

D) 17/25

Correct Answer: A

The quotient rule states that H'(x) = [g(x)f'(x) - f(x)g'(x)] / [g(x)]^2. To find H'(2), we substitute the given values: H'(2) = [g(2)f'(2) - f(2)g'(2)] / [g(2)]^2 = [(5)(-1) - (3)(4)] / (5)^2 = [-5 - 12] / 25 = -17 / 25.

Let f(x) = g(x) / h(x), where g and h are differentiable functions. Which of the following expressions represents f'(x)?

A) (g(x)h'(x) - h(x)g'(x)) / [h(x)]^2

B) (h(x)g'(x) - g(x)h'(x)) / h(x)

C) (h(x)g'(x) + g(x)h'(x)) / [h(x)]^2

D) (h(x)g'(x) - g(x)h'(x)) / [h(x)]^2

Correct Answer: D

This question is a direct application of the definition of the quotient rule. The rule for finding the derivative of a quotient of two functions, g(x) and h(x), is [h(x)g'(x) - g(x)h'(x)] / [h(x)]^2. Option A reverses the terms in the numerator. Option B does not square the denominator. Option C incorrectly uses addition, resembling the product rule.

What is the equation of the line tangent to the graph of y = (4x) / (x^2 + 1) at the point where x = 1?

A) y = 2

B) y - 2 = 4(x - 1)

C) y = 0

D) y - 2 = -2(x - 1)

Correct Answer: A

First, find the point of tangency by evaluating y at x = 1: y(1) = 4(1) / (1^2 + 1) = 4/2 = 2. The point is (1, 2). Next, find the slope of the tangent line by finding the derivative y' using the quotient rule: y' = [(x^2 + 1)(4) - (4x)(2x)] / (x^2 + 1)^2 = (4x^2 + 4 - 8x^2) / (x^2 + 1)^2 = (-4x^2 + 4) / (x^2 + 1)^2. Evaluate the slope at x = 1: m = y'(1) = (-4(1)^2 + 4) / (1^2 + 1)^2 = (-4 + 4) / 4 = 0. The slope is 0, indicating a horizontal line. Using the point-slope form y - y1 = m(x - x1), we get y - 2 = 0(x - 1), which simplifies to y = 2.

Which of the following is the derivative of the function f(x) = (x^3 - 2x) / 5?

A) 3x^2 - 2

B) (15x^2 - 10) / 25

C) (3x^2 - 2) / 5

D) (-5x^3 + 10x) / 25

Correct Answer: C

This function can be differentiated using the quotient rule, but it is simpler to treat it as a constant multiple. Rewrite f(x) as (1/5)(x^3 - 2x). Using the constant multiple rule, f'(x) = (1/5) * d/dx(x^3 - 2x) = (1/5)(3x^2 - 2), which is (3x^2 - 2) / 5. Alternatively, using the quotient rule with g(x) = x^3 - 2x and h(x) = 5, we have g'(x) = 3x^2 - 2 and h'(x) = 0. Then f'(x) = [(5)(3x^2 - 2) - (x^3 - 2x)(0)] / 5^2 = [5(3x^2 - 2)] / 25 = (3x^2 - 2) / 5.