Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![**Problem Statement:**
Find \( f'(x) \).
**Function Provided:**
\( f(x) = 2x^4 \ln x \)
**Solution Format:**
\( f'(x) = \underline{\hspace{1cm}} \)
**Explanation:**
The problem asks for the derivative of the given function \( f(x) = 2x^4 \ln x \). To find \( f'(x) \), use the product rule for differentiation, which states that if \( f(x) = u(x) \cdot v(x) \), then \( f'(x) = u'(x)v(x) + u(x)v'(x) \).
1. Let \( u(x) = 2x^4 \) and \( v(x) = \ln x \).
2. Differentiate \( u(x) \):
\[
u'(x) = \frac{d}{dx}[2x^4] = 8x^3
\]
3. Differentiate \( v(x) \):
\[
v'(x) = \frac{d}{dx}[\ln x] = \frac{1}{x}
\]
4. Apply the product rule:
\[
f'(x) = u'(x)v(x) + u(x)v'(x) = (8x^3)(\ln x) + (2x^4)\left(\frac{1}{x}\right)
\]
5. Simplify the expression:
\[
f'(x) = 8x^3 \ln x + 2x^3
\]
**Final Answer:**
\( f'(x) = 8x^3 \ln x + 2x^3 \)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbbe47654-762f-4eea-844b-6a49397579c0%2F6797d346-3b9b-41be-8691-6a5a7992ebe0%2F1vhiht_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find \( f'(x) \).
**Function Provided:**
\( f(x) = 2x^4 \ln x \)
**Solution Format:**
\( f'(x) = \underline{\hspace{1cm}} \)
**Explanation:**
The problem asks for the derivative of the given function \( f(x) = 2x^4 \ln x \). To find \( f'(x) \), use the product rule for differentiation, which states that if \( f(x) = u(x) \cdot v(x) \), then \( f'(x) = u'(x)v(x) + u(x)v'(x) \).
1. Let \( u(x) = 2x^4 \) and \( v(x) = \ln x \).
2. Differentiate \( u(x) \):
\[
u'(x) = \frac{d}{dx}[2x^4] = 8x^3
\]
3. Differentiate \( v(x) \):
\[
v'(x) = \frac{d}{dx}[\ln x] = \frac{1}{x}
\]
4. Apply the product rule:
\[
f'(x) = u'(x)v(x) + u(x)v'(x) = (8x^3)(\ln x) + (2x^4)\left(\frac{1}{x}\right)
\]
5. Simplify the expression:
\[
f'(x) = 8x^3 \ln x + 2x^3
\]
**Final Answer:**
\( f'(x) = 8x^3 \ln x + 2x^3 \)
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