If ƒ(x) = (7x − 2)² · (8x² + 8)ª, find ƒ'(x) using logarithmic differentiation

Calculus: Early Transcendentals
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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)}
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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