AP Calculus BC Practice Quiz: The Chain 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

If f(x) = (x^2 + 3)^5, then f'(x) =

A)5(x^2 + 3)^4B)5(2x)^4C)10x(x^2 + 3)^4D)5x(x^2 + 3)^4

All Questions (7)

If f(x) = (x^2 + 3)^5, then f'(x) =

A) 5(x^2 + 3)^4

B) 5(2x)^4

C) 10x(x^2 + 3)^4

D) 5x(x^2 + 3)^4

Correct Answer: C

This question requires calculating the derivative of a composition of functions, f(g(x)), where the outer function is u^5 and the inner function is g(x) = x^2 + 3. According to the chain rule, the derivative is f'(g(x)) * g'(x). The derivative of the outer function is 5(x^2 + 3)^4, and the derivative of the inner function is 2x. Multiplying them gives 10x(x^2 + 3)^4.

The derivative of the function y = sin(x^3) is:

A) cos(x^3)

B) 3x^2 cos(x^3)

C) 3x^2 sin(x^3)

D) -cos(x^3)

Correct Answer: B

To differentiate the composite function sin(x^3), the chain rule must be applied. The derivative of the outer function, sin(u), is cos(u). The derivative of the inner function, x^3, is 3x^2. The chain rule states we multiply the derivative of the outer function (evaluated at the inner function) by the derivative of the inner function, resulting in cos(x^3) * 3x^2, or 3x^2 cos(x^3).

The differentiation of which of the following functions requires the use of the chain rule?

A) f(x) = x^2 + cos(x)

B) f(x) = x^2 * cos(x)

C) f(x) = cos(x^2)

D) f(x) = x^2 / cos(x)

Correct Answer: C

The chain rule is used to differentiate composite functions (a function within a function). Option C, f(x) = cos(x^2), is a composition of the cosine function and the squaring function. Options A, B, and D require the sum, product, and quotient rules, respectively, but not the chain rule in their primary application.

Let f and g be differentiable functions. If h(x) = f(g(x)), what is h'(x)?

A) f'(x) * g'(x)

B) f'(g(x))

C) f(g'(x)) * g'(x)

D) f'(g(x)) * g'(x)

Correct Answer: D

This question asks for the definition of the chain rule for a composition of two differentiable functions. The chain rule states that the derivative of a composite function f(g(x)) is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. This corresponds to f'(g(x)) * g'(x).

Let f and g be differentiable functions with the following values: f(2)=3, f'(2)=4, g(2)=5, g'(2)=6. If h(x) = f(g(x)), what is the value of h'(2)?

A) 12

B) 20

C) 24

D) Cannot be determined

Correct Answer: D

To find h'(2), we use the chain rule: h'(x) = f'(g(x)) * g'(x). Plugging in x=2, we get h'(2) = f'(g(2)) * g'(2). From the given values, g(2)=5 and g'(2)=6. Substituting these in, we have h'(2) = f'(5) * 6. However, the value of f'(5) is not provided in the table. Therefore, the value of h'(2) cannot be determined from the given information.

If y = e^(cos(x)), then dy/dx =

A) e^(cos(x))

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

C) e^(-sin(x))

D) cos(x) * e^(cos(x)-1)

Correct Answer: B

This problem requires the differentiation of a composite function, e^u, where u = cos(x). Using the chain rule, we differentiate the outer function, e^u, which is e^u, and multiply it by the derivative of the inner function, cos(x), which is -sin(x). The result is e^(cos(x)) * (-sin(x)), or -sin(x) * e^(cos(x)).

Let g be a differentiable function such that g(1) = 4 and g'(1) = 2. If h(x) = sqrt(g(x)), what is the value of h'(1)?

A) 1/4

B) 1/2

C) 1

D) 4

Correct Answer: B

The function h(x) is a composition of f(u) = sqrt(u) = u^(1/2) and u = g(x). By the chain rule, h'(x) = f'(g(x)) * g'(x). The derivative of f(u) is f'(u) = (1/2)u^(-1/2) = 1/(2*sqrt(u)). Therefore, h'(x) = [1 / (2*sqrt(g(x)))] * g'(x). To find h'(1), we substitute x=1: h'(1) = [1 / (2*sqrt(g(1)))] * g'(1). Using the given values g(1)=4 and g'(1)=2, we get h'(1) = [1 / (2*sqrt(4))] * 2 = [1 / (2*2)] * 2 = (1/4) * 2 = 1/2.