AP Calculus AB Practice Quiz: Differentiating Inverse Functions

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

Which of the following is the derivative of the inverse trigonometric function f(x) = arctan(x)?

A)1 / (1 + x^2)B)1 / sqrt(1 - x^2)C)-1 / (1 + x^2)D)-1 / sqrt(1 - x^2)

All Questions (7)

Which of the following is the derivative of the inverse trigonometric function f(x) = arctan(x)?

A) 1 / (1 + x^2)

B) 1 / sqrt(1 - x^2)

C) -1 / (1 + x^2)

D) -1 / sqrt(1 - x^2)

Correct Answer: A

This question tests the ability to calculate the derivative of a standard inverse trigonometric function. The derivative of arctan(x) with respect to x is a known formula: d/dx(arctan(x)) = 1 / (1 + x^2).

Let f(x) = arcsin(x^2). What is f'(x)?

A) 2x / sqrt(1 - x^4)

B) 1 / sqrt(1 - x^4)

C) 2x / (1 + x^4)

D) 2x / sqrt(1 - x^2)

Correct Answer: A

This question requires using the chain rule to calculate the derivative of a composite inverse trigonometric function. Let u = x^2, so du/dx = 2x. The derivative of arcsin(u) is 1/sqrt(1 - u^2). By the chain rule, f'(x) = (d/du(arcsin(u))) * (du/dx) = (1/sqrt(1 - (x^2)^2)) * (2x) = 2x / sqrt(1 - x^4).

Let f be a differentiable function such that f(3) = 5 and f'(3) = 2. If g is the inverse function of f, what is the value of g'(5)?

A) 2

B) 1/2

C) 1/5

D) -2

Correct Answer: B

The derivative of an inverse function g at a point a is given by the formula g'(a) = 1 / f'(g(a)). We are asked for g'(5). From the definition of an inverse function, if f(3) = 5, then g(5) = 3. Therefore, g'(5) = 1 / f'(g(5)) = 1 / f'(3). Since we are given that f'(3) = 2, the answer is 1/2.

The derivative of an inverse function can be found using which combination of mathematical principles, provided the derivative exists?

A) The product rule and the definition of a limit

B) The quotient rule and implicit differentiation

C) The chain rule and the definition of an inverse function

D) The power rule and L'Hôpital's Rule

Correct Answer: C

The provided content explicitly states that 'The chain rule and definition of an inverse function can be used to find the derivative of an inverse function'. This is demonstrated by differentiating the identity f(g(x)) = x, which yields f'(g(x)) * g'(x) = 1, and then solving for g'(x).

Let f(x) = x^3 + 2x + 1. If g is the inverse function of f, what is the value of g'(4)?

A) 1/5

B) 1/14

C) 5

D) 1/31

Correct Answer: A

To find g'(4), we use the formula g'(4) = 1 / f'(g(4)). First, we must find the value of g(4). This is the value 'a' for which f(a) = 4. So, we solve a^3 + 2a + 1 = 4, which simplifies to a^3 + 2a - 3 = 0. By inspection, a = 1 is the solution (1^3 + 2(1) - 3 = 0). Thus, g(4) = 1. Next, we find the derivative of f(x): f'(x) = 3x^2 + 2. Now we evaluate f'(g(4)), which is f'(1). f'(1) = 3(1)^2 + 2 = 5. Therefore, g'(4) = 1 / f'(1) = 1/5.

Which of the following is the derivative of the inverse trigonometric function f(x) = arccos(x)?

A) 1 / sqrt(1 - x^2)

B) -1 / sqrt(1 - x^2)

C) 1 / (1 + x^2)

D) -1 / (1 + x^2)

Correct Answer: B

This question tests the ability to calculate the derivative of a standard inverse trigonometric function. The derivative of arccos(x) with respect to x is a known formula: d/dx(arccos(x)) = -1 / sqrt(1 - x^2). A common mistake is to forget the negative sign.

Let f(x) = e^x + x and let g(x) be the inverse function of f. What is the slope of the tangent line to the graph of y = g(x) at the point where x = 1?

A) 1/2

B) 1

C) 2

D) 1 / (e + 1)

Correct Answer: A

The slope of the tangent line to g(x) at x=1 is g'(1). We use the formula g'(1) = 1 / f'(g(1)). First, we find g(1) by solving f(a) = 1, which is e^a + a = 1. By inspection, a = 0 is the solution (e^0 + 0 = 1). So, g(1) = 0. Next, we find the derivative of f(x): f'(x) = e^x + 1. Now, we evaluate f'(g(1)), which is f'(0). f'(0) = e^0 + 1 = 1 + 1 = 2. Therefore, the slope g'(1) is 1 / f'(0) = 1/2.