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 \( f(x) = (3x - 4)^3 \cdot (8x^2 + 6)^4 \), find \( f'(x) \) using logarithmic differentiation.
### Instructions
To solve this problem, you will apply logarithmic differentiation, a technique that utilizes the properties of logarithms to simplify the differentiation of complex functions, especially those involving products or powers of functions.
### Steps:
1. **Take the natural logarithm of both sides:**
Start by letting \( y = f(x) \), which means:
\[
y = (3x - 4)^3 \cdot (8x^2 + 6)^4
\]
Then take the natural logarithm:
\[
\ln(y) = \ln((3x - 4)^3 \cdot (8x^2 + 6)^4)
\]
2. **Expand using logarithm properties:**
Use the property that \(\ln(ab) = \ln(a) + \ln(b)\) and \(\ln(a^b) = b\ln(a)\):
\[
\ln(y) = 3\ln(3x - 4) + 4\ln(8x^2 + 6)
\]
3. **Differentiate both sides:**
Differentiate implicitly with respect to \(x\):
\[
\frac{1}{y} \frac{dy}{dx} = 3\left(\frac{1}{3x - 4}\right)(3) + 4\left(\frac{1}{8x^2 + 6}\right)(16x)
\]
4. **Simplify the derivatives:**
\[
\frac{1}{y} \frac{dy}{dx} = \frac{9}{3x - 4} + \frac{64x}{8x^2 + 6}
\]
5. **Solve for \( \frac{dy}{dx} \) (i.e., \( f'(x) \)):**
Multiply through by \( y \) (which is \( (3x - 4)^3 \cdot (8x^2 + 6)^4 \)) to solve for \( \frac{dy}{dx} \):
\[
f'(x)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F08bb2a94-da17-43f2-90ab-4ad444df37b3%2F92054760-94e8-4f62-8c5e-53f48960eca0%2Fojk2bka_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
If \( f(x) = (3x - 4)^3 \cdot (8x^2 + 6)^4 \), find \( f'(x) \) using logarithmic differentiation.
### Instructions
To solve this problem, you will apply logarithmic differentiation, a technique that utilizes the properties of logarithms to simplify the differentiation of complex functions, especially those involving products or powers of functions.
### Steps:
1. **Take the natural logarithm of both sides:**
Start by letting \( y = f(x) \), which means:
\[
y = (3x - 4)^3 \cdot (8x^2 + 6)^4
\]
Then take the natural logarithm:
\[
\ln(y) = \ln((3x - 4)^3 \cdot (8x^2 + 6)^4)
\]
2. **Expand using logarithm properties:**
Use the property that \(\ln(ab) = \ln(a) + \ln(b)\) and \(\ln(a^b) = b\ln(a)\):
\[
\ln(y) = 3\ln(3x - 4) + 4\ln(8x^2 + 6)
\]
3. **Differentiate both sides:**
Differentiate implicitly with respect to \(x\):
\[
\frac{1}{y} \frac{dy}{dx} = 3\left(\frac{1}{3x - 4}\right)(3) + 4\left(\frac{1}{8x^2 + 6}\right)(16x)
\]
4. **Simplify the derivatives:**
\[
\frac{1}{y} \frac{dy}{dx} = \frac{9}{3x - 4} + \frac{64x}{8x^2 + 6}
\]
5. **Solve for \( \frac{dy}{dx} \) (i.e., \( f'(x) \)):**
Multiply through by \( y \) (which is \( (3x - 4)^3 \cdot (8x^2 + 6)^4 \)) to solve for \( \frac{dy}{dx} \):
\[
f'(x)
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