AP Calculus AB 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
All Questions (7)
A) 5(x^3 - 4x)^4
B) 5(3x^2 - 4)^4
C) 5(x^3 - 4x)^4 (3x^2 - 4)
D) 15x^2(x^3 - 4x)^4
Correct Answer: C
This problem requires the chain rule for a composite function. Let the outer function be u^5 and the inner function be u = x^3 - 4x. The derivative of the outer function is 5u^4. The derivative of the inner function is 3x^2 - 4. According to the chain rule, the derivative is the product of the derivative of the outer function (with the inner function plugged in) and the derivative of the inner function: 5(x^3 - 4x)^4 * (3x^2 - 4).
A) cos(2x^4)
B) 8x^3 cos(2x^4)
C) -8x^3 cos(2x^4)
D) cos(8x^3)
Correct Answer: B
To differentiate the composite function sin(2x^4), we use the chain rule. The outer function is sin(u) and its derivative is cos(u). The inner function is u = 2x^4 and its derivative is 8x^3. The chain rule states we multiply the derivative of the outer function by the derivative of the inner function: cos(2x^4) * 8x^3, which simplifies to 8x^3 cos(2x^4).
A) f'(g(x)) * g'(x)
B) f'(x) * g'(x)
C) f(g'(x))
D) f'(g(x)) + g'(x)
Correct Answer: A
The chain rule provides a method for differentiating composite functions. For a function h(x) = f(g(x)), the rule states that the derivative h'(x) is the derivative of the outer function f, evaluated at the inner function g(x), multiplied by the derivative of the inner function g'(x). This is formally written as h'(x) = f'(g(x)) * g'(x).
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 is a composition of functions where the outer function is e^u and the inner function is u = cos(x). The derivative of e^u is e^u. The derivative of cos(x) is -sin(x). Applying the chain rule, we multiply the derivative of the outer function by the derivative of the inner function: e^(cos(x)) * (-sin(x)), which is -sin(x)e^(cos(x)).
A) sec^2(x) / tan(x)
B) 1 / tan(x)
C) cot(x) / sec^2(x)
D) 1 / (x * sec^2(x))
Correct Answer: A
This problem requires the chain rule. The outer function is ln(u) and the inner function is u = tan(x). The derivative of the outer function is 1/u. The derivative of the inner function is sec^2(x). By the chain rule, the derivative is (1/tan(x)) * sec^2(x), which equals sec^2(x) / tan(x). This can also be simplified to csc(x)sec(x), but the unsimplified form is given as an option.
A) -12 cos^2(4x) sin(4x)
B) -3 cos^2(4x) sin(4x)
C) 12 cos^2(4x)
D) -4 sin^3(4x)
Correct Answer: A
This problem involves a nested composition, requiring the chain rule to be applied twice. The function can be written as y = (cos(4x))^3. The outermost function is u^3, the middle function is cos(v), and the innermost function is 4x. The derivatives are 3u^2, -sin(v), and 4, respectively. Applying the chain rule: dy/dx = 3(cos(4x))^2 * (-sin(4x)) * 4 = -12 cos^2(4x) sin(4x).
A) 1 / (2 * sqrt(x^2 + 9))
B) x / sqrt(x^2 + 9)
C) 2x * sqrt(x^2 + 9)
D) x / (2 * sqrt(x^2 + 9))
Correct Answer: B
The function can be written as f(x) = (x^2 + 9)^(1/2). This is a composite function where the outer function is u^(1/2) and the inner function is u = x^2 + 9. The derivative of the outer function is (1/2)u^(-1/2). The derivative of the inner function is 2x. Using the chain rule, f'(x) = (1/2)(x^2 + 9)^(-1/2) * (2x). This simplifies to (2x) / (2 * sqrt(x^2 + 9)), and further simplifies to x / sqrt(x^2 + 9).