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...
Related questions
Question
![**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) \(12x + 16\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0782be39-c258-415f-a3e1-efee620e7e00%2F37ff5788-00f2-4f45-aa0e-4860b767e71a%2Fzg82bsw_processed.jpeg&w=3840&q=75)
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) \(12x + 16\).
Expert Solution
Step 1
Given query is to find the Indicated derivative.
Step by step
Solved in 2 steps with 1 images
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