Evaluate (2x + 4)(3x + 2) d dx A) 6 C) 12x + 16 B) 6x² + 16x + 8 D) 30x² + 43x + 8

Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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**Problem:** Evaluate \(\frac{d}{dx}(2x + 4)(3x + 2)\).

**Options:**

A) 6

B) \(6x^2 + 16x + 8\)

C) \(12x + 16\)

D) \(30x^2 + 43x + 8\)

**Explanation:**

To solve this problem, apply the product rule of differentiation, which states:

If \(u(x) = f(x) \cdot g(x)\), then \( \frac{d}{dx}[u(x)] = f'(x)g(x) + f(x)g'(x) \).

Let \(f(x) = 2x + 4\) and \(g(x) = 3x + 2\).

1. Compute \(f'(x)\): The derivative of \(2x + 4\) is 2.
2. Compute \(g'(x)\): The derivative of \(3x + 2\) is 3.
3. Apply the product rule:

\[
\frac{d}{dx}[(2x + 4)(3x + 2)] = (2)(3x + 2) + (2x + 4)(3)
\]

Evaluate:

\[
= (2)(3x) + (2)(2) + (2x)(3) + (4)(3) 
= 6x + 4 + 6x + 12
= 12x + 16
\]

Therefore, the correct option is C) \(12x + 16\).
Transcribed Image Text:**Problem:** Evaluate \(\frac{d}{dx}(2x + 4)(3x + 2)\). **Options:** A) 6 B) \(6x^2 + 16x + 8\) C) \(12x + 16\) D) \(30x^2 + 43x + 8\) **Explanation:** To solve this problem, apply the product rule of differentiation, which states: If \(u(x) = f(x) \cdot g(x)\), then \( \frac{d}{dx}[u(x)] = f'(x)g(x) + f(x)g'(x) \). Let \(f(x) = 2x + 4\) and \(g(x) = 3x + 2\). 1. Compute \(f'(x)\): The derivative of \(2x + 4\) is 2. 2. Compute \(g'(x)\): The derivative of \(3x + 2\) is 3. 3. Apply the product rule: \[ \frac{d}{dx}[(2x + 4)(3x + 2)] = (2)(3x + 2) + (2x + 4)(3) \] Evaluate: \[ = (2)(3x) + (2)(2) + (2x)(3) + (4)(3) = 6x + 4 + 6x + 12 = 12x + 16 \] Therefore, the correct option is C) \(12x + 16\).
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