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 Statement:**
Find the following derivative:
\[
\frac{d}{dx} \left( (x^3 + 4) \ln x \right)
\]
---
**Solution:**
To solve this problem, you will need to use the *product rule* for differentiation. The product rule states that if you have a product of two functions, say \( u(x) \) and \( v(x) \), their derivative is:
\[
\frac{d}{dx}[u(x) \cdot v(x)] = u'(x) v(x) + u(x) v'(x)
\]
In this case, identify:
- \( u(x) = x^3 + 4 \) and \( u'(x) = 3x^2 \)
- \( v(x) = \ln x \) and \( v'(x) = \frac{1}{x} \)
Apply the product rule:
\[
\frac{d}{dx} \left( (x^3 + 4) \ln x \right) = (3x^2) \ln x + (x^3 + 4) \cdot \frac{1}{x}
\]
Simplifying the expression:
\[
= 3x^2 \ln x + x^2 + \frac{4}{x}
\]
Thus, the derivative of \( (x^3 + 4) \ln x \) is:
\[
3x^2 \ln x + x^2 + \frac{4}{x}
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0f3dd2f2-0071-4cf8-a607-c0ff1b3f883f%2F11ffb4bb-2cb8-4ed1-a52f-78aa5dbdab4c%2Fne6l44_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the following derivative:
\[
\frac{d}{dx} \left( (x^3 + 4) \ln x \right)
\]
---
**Solution:**
To solve this problem, you will need to use the *product rule* for differentiation. The product rule states that if you have a product of two functions, say \( u(x) \) and \( v(x) \), their derivative is:
\[
\frac{d}{dx}[u(x) \cdot v(x)] = u'(x) v(x) + u(x) v'(x)
\]
In this case, identify:
- \( u(x) = x^3 + 4 \) and \( u'(x) = 3x^2 \)
- \( v(x) = \ln x \) and \( v'(x) = \frac{1}{x} \)
Apply the product rule:
\[
\frac{d}{dx} \left( (x^3 + 4) \ln x \right) = (3x^2) \ln x + (x^3 + 4) \cdot \frac{1}{x}
\]
Simplifying the expression:
\[
= 3x^2 \ln x + x^2 + \frac{4}{x}
\]
Thus, the derivative of \( (x^3 + 4) \ln x \) is:
\[
3x^2 \ln x + x^2 + \frac{4}{x}
\]
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