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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![**Logarithmic Differentiation Example**
Use logarithmic differentiation to find the derivative of the function.
Given:
\[ y = (\ln(x))^{\cos(8x)} \]
To find \[ y' \], we differentiate using logarithmic differentiation:
\[ y' = -\frac{x \ln(x)^{\cos(8x)} \ln(\ln^8(x)) \sin(8x) - \ln(x)^{\cos(8x) - 1} \cos(8x)}{x} \]
**Explanation:**
In the equation, logarithmic differentiation is used to differentiate a function where the variable \( x \) is both the base and the exponent. Here, \( \ln(x) \) is raised to the power of \( \cos(8x) \).
The derivative \( y' \) incorporates several components:
- **Numerator:**
- \( x \ln(x)^{\cos(8x)} \): Represents the function raised to the power.
- \( \ln(\ln^8(x)) \): The natural logarithm of \( \ln(x) \) raised to the 8th power.
- \( \sin(8x) \): The sine of \( 8x \).
- Minus \( \ln(x)^{\cos(8x) - 1} \cos(8x) \): Adjusts for the power using the chain rule.
- **Denominator:**
- \( x \): Reduces the expression, simplifying the differentiation.
Logarithmic differentiation is especially useful for functions with variable exponents, allowing the decomposition of the function into manageable parts for differentiation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6e533fec-51fd-4870-9b3f-7eb1d9840991%2F512a2e3b-25eb-4b61-96e8-d590929feca0%2Fx3qcel_processed.png&w=3840&q=75)
Transcribed Image Text:**Logarithmic Differentiation Example**
Use logarithmic differentiation to find the derivative of the function.
Given:
\[ y = (\ln(x))^{\cos(8x)} \]
To find \[ y' \], we differentiate using logarithmic differentiation:
\[ y' = -\frac{x \ln(x)^{\cos(8x)} \ln(\ln^8(x)) \sin(8x) - \ln(x)^{\cos(8x) - 1} \cos(8x)}{x} \]
**Explanation:**
In the equation, logarithmic differentiation is used to differentiate a function where the variable \( x \) is both the base and the exponent. Here, \( \ln(x) \) is raised to the power of \( \cos(8x) \).
The derivative \( y' \) incorporates several components:
- **Numerator:**
- \( x \ln(x)^{\cos(8x)} \): Represents the function raised to the power.
- \( \ln(\ln^8(x)) \): The natural logarithm of \( \ln(x) \) raised to the 8th power.
- \( \sin(8x) \): The sine of \( 8x \).
- Minus \( \ln(x)^{\cos(8x) - 1} \cos(8x) \): Adjusts for the power using the chain rule.
- **Denominator:**
- \( x \): Reduces the expression, simplifying the differentiation.
Logarithmic differentiation is especially useful for functions with variable exponents, allowing the decomposition of the function into manageable parts for differentiation.
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