AP Calculus BC : 3.5 Selecting Procedures for Calculating Derivatives- Exam Style questions with Answer- MCQ

Question

Which of the following expressions can be differentiated using the product rule?

A. \( \arcsin(\cos x) \)

B. \( \sin x (\arccos x) \)

C. \( e^x + \arctan x \)

D. \( (12x^2 + 3x – 6)^e \)

▶️ Answer/Explanation

Solution

The product rule applies to \( u(x) \cdot v(x) \).

A. \( \arcsin(\cos x) \): Composition, use chain rule.

B. \( \sin x \cdot \arccos x \): Product, use product rule.

C. \( e^x + \arctan x \): Sum, use sum rule.

D. \( (12x^2 + 3x – 6)^e \): Power, use power rule with chain rule.

Only B uses the product rule.

✅ Answer: B)

Question

Which of the following requires the use of implicit differentiation to find \( \frac{dy}{dx} \)?

A) \( 2y + 3x^2 – x = 5 \)

B) \( y = e^3 + x + x^3 \)

C) \( y = e^{y + x} + x^3 \)

D) \( y = \frac{x^4 + 3}{4x^3 – 2} \)

▶️ Answer/Explanation

Solution

A. Solve: \( y = \frac{5 – 3x^2 + x}{2} \). Direct differentiation applies.

B. \( y = e^3 + x + x^3 \): Direct differentiation applies.

C. \( y = e^{y + x} + x^3 \): \( y \) on both sides; implicit differentiation is required.

D. \( y = \frac{x^4 + 3}{4x^3 – 2} \): Use quotient rule, not implicit.

Only C requires implicit differentiation.

✅ Answer: C)

Question

For which of the following functions would the quotient rule be considered the best method for finding the derivative?

A) \( y = (x^3 + x)^{-2} \)

B) \( y = \frac{x^3 + x}{x} \)

C) \( y = \cos^{-1}(x^3 + x) \)

D) \( y = \frac{\cos(x^3 + x)}{x^3 + x} \)

▶️ Answer/Explanation

Solution

A. \( (x^3 + x)^{-2} \): Use chain rule, not quotient rule.

B. \( \frac{x^3 + x}{x} = x^2 + 1 \): Simplify and differentiate, not quotient rule.

C. \( \cos^{-1}(x^3 + x) \): Use chain rule, not quotient rule.

D. \( \frac{\cos(x^3 + x)}{x^3 + x} \): Use quotient rule as best method.

Only D requires the quotient rule.

✅ Answer: D)

Question

For \( 0 < x < \frac{\pi}{2} \), if \( y = (\sin x)^x \), then \( \frac{dy}{dx} \) is

(A) \( x \ln (\sin x) \)

(B) \( (\sin x)^x \cot x \)

(D) \( (\sin x)^x (x \cos x + \sin x) \)

(E) \( (\sin x)^x (x \cot x + \ln (\sin x)) \)

▶️ Answer/Explanation

Solution

Use logarithmic differentiation: \( \ln y = x \ln (\sin x) \).

Differentiate: \( \frac{1}{y} \frac{dy}{dx} = x \cot x + \ln (\sin x) \).

Solve: \( \frac{dy}{dx} = (\sin x)^x (x \cot x + \ln (\sin x)) \).

✅ Answer: E)

AP Calculus BC : 3.5 Selecting Procedures for Calculating Derivatives- Exam Style questions with Answer- MCQ

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