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AP Calculus BC Practice Quiz: Selecting Procedures for Calculating Derivatives

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

To find the derivative of the function h(x) = x² * sin(x), which differentiation procedure is most appropriate to apply first?

All Questions (7)

To find the derivative of the function h(x) = x² * sin(x), which differentiation procedure is most appropriate to apply first?

A) Chain Rule

B) Product Rule

C) Quotient Rule

D) Implicit Differentiation

Correct Answer: B

The function h(x) is presented as the product of two distinct functions, f(x) = x² and g(x) = sin(x). Therefore, the Product Rule is the correct procedure for finding its derivative.

Which procedure is required to find the derivative of g(x) = (x³ + 4) / eˣ?

A) Quotient Rule

B) Product Rule

C) Logarithmic Differentiation

D) Chain Rule

Correct Answer: A

The function g(x) is structured as a fraction with a function in the numerator and a function in the denominator. The most direct procedure for differentiating such a function is the Quotient Rule.

Calculating the derivative of f(x) = cos(ln(x²)) requires the primary application of which differentiation rule?

A) Product Rule

B) Quotient Rule

C) Chain Rule

D) Power Rule

Correct Answer: C

The function f(x) is a composition of functions (a function within a function within a function). The procedure for differentiating composite functions is the Chain Rule, which would need to be applied multiple times.

To find dy/dx for the equation x² + y² = 25, which differentiation procedure must be used?

A) Logarithmic Differentiation

B) Parametric Differentiation

C) Differentiation of an Inverse

D) Implicit Differentiation

Correct Answer: D

The equation does not explicitly define y as a function of x. To find the derivative of y with respect to x, one must use implicit differentiation, treating y as a function of x and applying the Chain Rule accordingly.

What is the most effective procedure for finding the derivative of the function y = x^(sin(x))?

A) Power Rule

B) Logarithmic Differentiation

C) Product Rule

D) Quotient Rule

Correct Answer: B

This function has a variable in both the base and the exponent. The Power Rule cannot be used. The most effective procedure is Logarithmic Differentiation, which involves taking the natural logarithm of both sides before differentiating.

Finding the derivative of f(x) = e^(3x) / (x² + 1) requires which combination of differentiation procedures?

A) Product Rule and Chain Rule

B) Quotient Rule and Chain Rule

C) Implicit Differentiation and Product Rule

D) Logarithmic Differentiation and Power Rule

Correct Answer: B

The overall structure of the function is a quotient, making the Quotient Rule the primary procedure. The numerator, e^(3x), is a composite function, which requires the Chain Rule for its differentiation. Therefore, both rules are necessary.

Given a differentiable, invertible function f(x) and its inverse g(x) = f⁻¹(x), which procedure is used to find the value of g'(a) for some constant a?

A) Applying the Quotient Rule to 1/f(x).

B) Using the formula for the derivative of an inverse function, g'(a) = 1 / f'(g(a)).

C) Applying the Product Rule to f(x) * g(x).

D) Using implicit differentiation on the equation f(x) + g(x) = 0.

Correct Answer: B

The standard procedure for finding the derivative of an inverse function at a point is to use the specific formula derived for this purpose: g'(a) = 1 / f'(g(a)). This relates the derivative of the inverse to the derivative of the original function.