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AP Calculus BC 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

According to the provided principles, which combination of mathematical concepts is used to find the derivative of an inverse function?

All Questions (7)

According to the provided principles, which combination of mathematical concepts is used to find the derivative of an inverse function?

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'.

Let f be a differentiable function such that f(3) = 8 and f'(3) = 5. If g is the inverse function of f, what is the value of g'(8)?

A) 5

B) 1/5

C) 1/8

D) -5

Correct Answer: B

The rule for the derivative of an inverse function is g'(y) = 1 / f'(x), where y = f(x). Here, y = 8 and x = 3. Therefore, g'(8) = 1 / f'(3) = 1/5.

What is the derivative of y = arctan(2x) with respect to x?

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

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

C) 2 / (1 + x^2)

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

Correct Answer: A

This problem requires calculating the derivative of an inverse trigonometric function using the chain rule. The derivative of arctan(u) is u' / (1 + u^2). Here, u = 2x, so u' = 2. Substituting these into the formula gives 2 / (1 + (2x)^2) = 2 / (1 + 4x^2).

The table below gives values of a differentiable function f and its derivative f' at selected values of x. If g(x) = f⁻¹(x), what is the value of g'(5)? | x | f(x) | f'(x) | |---|---|---| | 2 | 5 | -3 | | 5 | 8 | 4 |

A) 1/4

B) -1/3

C) 4

D) 1/8

Correct Answer: B

To find g'(5), we use the formula g'(y) = 1 / f'(x) where y = f(x). We need to find the x-value for which f(x) = 5. From the table, we see that f(2) = 5. Therefore, g'(5) = 1 / f'(2). From the table, f'(2) = -3. So, g'(5) = 1/(-3) = -1/3.

What is a necessary condition for finding the derivative of an inverse function f⁻¹ at a point?

A) The original function f must be a polynomial.

B) The original function f must be continuous.

C) The derivative of the original function f must exist and be non-zero at the corresponding point.

D) The inverse function f⁻¹ must be an inverse trigonometric function.

Correct Answer: C

The provided content states that the derivative of an inverse function can be found 'provided the derivative exists'. The formula for the derivative of an inverse, (f⁻¹)'(y) = 1/f'(x), also implies that f'(x) cannot be zero.

Let f(x) = x⁵ + 2x + 1. If g is the inverse function of f, what is the value of g'(4)?

A) 1/7

B) 1/81

C) 7

D) 1/163

Correct Answer: A

We need to find g'(4). First, find the value of x such that f(x) = 4. By inspection, x⁵ + 2x + 1 = 4 when x = 1. Next, find the derivative of f(x): f'(x) = 5x⁴ + 2. Now, evaluate f'(1) = 5(1)⁴ + 2 = 7. The derivative of the inverse is the reciprocal of this value, so g'(4) = 1 / f'(1) = 1/7.

Which of the following represents the derivative of arcsin(x) with respect to x?

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

B) 1 / (1 + x^2)

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

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

Correct Answer: C

This is a direct application of the rule for differentiating inverse trigonometric functions. The derivative of arcsin(x) is a standard result: d/dx(arcsin(x)) = 1 / sqrt(1 - x^2).