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 Statement
If \( f(x) = (7x - 2)^2 \cdot (8x^2 + 8)^{\frac{4}{3}} \), find \( f'(x) \) using logarithmic differentiation.
### Explanation
In this problem, we are given a function \( f(x) \) which is a product of two terms: \( (7x - 2)^2 \) and \( (8x^2 + 8)^{\frac{4}{3}} \).
To find the derivative \( f'(x) \) using logarithmic differentiation, we follow these steps:
1. **Take the natural logarithm of both sides of the given function**:
\[
\ln(f(x)) = \ln\left((7x - 2)^2 \cdot (8x^2 + 8)^{\frac{4}{3}}\right)
\]
2. **Use the properties of logarithms to simplify**:
\[
\ln(f(x)) = \ln((7x - 2)^2) + \ln((8x^2 + 8)^{\frac{4}{3}})
\]
\[
\ln(f(x)) = 2 \ln(7x - 2) + \frac{4}{3} \ln(8x^2 + 8)
\]
3. **Differentiate both sides with respect to \( x \)**, using the chain rule:
\[
\frac{d}{dx} \left[ \ln(f(x)) \right] = \frac{d}{dx} \left[ 2 \ln(7x - 2) + \frac{4}{3} \ln(8x^2 + 8) \right]
\]
\[
\frac{f'(x)}{f(x)} = 2 \cdot \frac{1}{7x - 2} \cdot 7 + \frac{4}{3} \cdot \frac{1}{8x^2 + 8} \cdot 16x
\]
\[
\frac{f'(x)}{f(x)} = \frac{14}{7x - 2} + \frac{64x}{3(8x^2 + 8)}](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F621ac9f8-2dbb-4715-ab7b-42c3ebc90b44%2F35fe7cbf-50ca-4686-8b78-a075db28bb40%2Ffuuvem_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement
If \( f(x) = (7x - 2)^2 \cdot (8x^2 + 8)^{\frac{4}{3}} \), find \( f'(x) \) using logarithmic differentiation.
### Explanation
In this problem, we are given a function \( f(x) \) which is a product of two terms: \( (7x - 2)^2 \) and \( (8x^2 + 8)^{\frac{4}{3}} \).
To find the derivative \( f'(x) \) using logarithmic differentiation, we follow these steps:
1. **Take the natural logarithm of both sides of the given function**:
\[
\ln(f(x)) = \ln\left((7x - 2)^2 \cdot (8x^2 + 8)^{\frac{4}{3}}\right)
\]
2. **Use the properties of logarithms to simplify**:
\[
\ln(f(x)) = \ln((7x - 2)^2) + \ln((8x^2 + 8)^{\frac{4}{3}})
\]
\[
\ln(f(x)) = 2 \ln(7x - 2) + \frac{4}{3} \ln(8x^2 + 8)
\]
3. **Differentiate both sides with respect to \( x \)**, using the chain rule:
\[
\frac{d}{dx} \left[ \ln(f(x)) \right] = \frac{d}{dx} \left[ 2 \ln(7x - 2) + \frac{4}{3} \ln(8x^2 + 8) \right]
\]
\[
\frac{f'(x)}{f(x)} = 2 \cdot \frac{1}{7x - 2} \cdot 7 + \frac{4}{3} \cdot \frac{1}{8x^2 + 8} \cdot 16x
\]
\[
\frac{f'(x)}{f(x)} = \frac{14}{7x - 2} + \frac{64x}{3(8x^2 + 8)}
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