Apentix/AP Calculus AB/Unit 2: Differentiation: Definition and Basic Derivative Rules/Question
AP Calculus ABhardmcq1 pt

If f(x) = (3x² + 2x)/(x³ - 1), then f'(x) = ?

A.B) (6x + 2)(x³ - 1) + (3x² + 2x)(3x²)/(x³ - 1)^2
B.A) (6x + 2)(x³ - 1) - (3x² + 2x)(3x²)/(x³ - 1)^2
C.C) (6x + 2)(x³ - 1) - (3x² + 2x)(3x²)/(x³ - 1)
D.D) (6x + 2)(x³ - 1) + (3x² + 2x)(3x²)/(x³ - 1)

Explanation

Core Concept

This requires the quotient rule. If f(x) = g(x)/h(x), then f'(x) = [g'(x)h(x) - g(x)h'(x)]/[h(x)]^2. Here, g(x) = 3x² + 2x, so g'(x) = 6x + 2. And h(x) = x³ - 1, so h'(x) = 3x². Therefore, f'(x) = [(6x + 2)(x³ - 1) - (3x² + 2x)(3x²)]/(x³ - 1)^2, which is option A. Option B is incorrect because it has a plus sign instead of a minus sign between the terms. Option C is incorrect because it doesn't square the denominator. Option D is incorrect because it has a plus sign and doesn't square the denominator.

Correct Answer

BA) (6x + 2)(x³ - 1) - (3x² + 2x)(3x²)/(x³ - 1)^2

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