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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Determine g’(x)
![**Problem Statement:**
Determine \( g'(x) \) if \( g(x) = \ln \left( \frac{x^2 + 5x + 8}{x^3 + 5x + 8} \right) \).
**Solution Overview:**
To find the derivative \( g'(x) \), we will use the chain rule and the properties of logarithmic differentiation. The function \( g(x) \) is a natural logarithm of a fraction, specifically \( \ln(u(x)) \) where:
\[ u(x) = \frac{x^2 + 5x + 8}{x^3 + 5x + 8} \]
### Steps:
1. **Differentiate the logarithmic function:**
Using the derivative of the natural logarithm, \((\ln(u))' = \frac{1}{u} \cdot u'\).
2. **Apply the quotient rule to find \( u'(x) \):**
For a function \( \frac{f(x)}{h(x)} \), the quotient rule states:
\[
\left(\frac{f}{h}\right)' = \frac{f'h - fh'}{h^2}
\]
3. **Determine \( f(x) = x^2 + 5x + 8 \) and \( h(x) = x^3 + 5x + 8 \):**
- \( f'(x) = 2x + 5 \)
- \( h'(x) = 3x^2 + 5 \)
4. **Apply the quotient rule:**
\[
u'(x) = \frac{(2x + 5)(x^3 + 5x + 8) - (x^2 + 5x + 8)(3x^2 + 5)}{(x^3 + 5x + 8)^2}
\]
5. **Plug \( u \) and \( u' \) into the derivative of the logarithmic function:**
\[
g'(x) = \frac{1}{u(x)} \cdot u'(x)
\]
Simplify this expression for the final result.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4cbe10c6-7416-4dbe-aac3-86d00105e150%2F2d0f6701-7d56-4e8c-828c-dcbd2388acf5%2Fifw0to_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Determine \( g'(x) \) if \( g(x) = \ln \left( \frac{x^2 + 5x + 8}{x^3 + 5x + 8} \right) \).
**Solution Overview:**
To find the derivative \( g'(x) \), we will use the chain rule and the properties of logarithmic differentiation. The function \( g(x) \) is a natural logarithm of a fraction, specifically \( \ln(u(x)) \) where:
\[ u(x) = \frac{x^2 + 5x + 8}{x^3 + 5x + 8} \]
### Steps:
1. **Differentiate the logarithmic function:**
Using the derivative of the natural logarithm, \((\ln(u))' = \frac{1}{u} \cdot u'\).
2. **Apply the quotient rule to find \( u'(x) \):**
For a function \( \frac{f(x)}{h(x)} \), the quotient rule states:
\[
\left(\frac{f}{h}\right)' = \frac{f'h - fh'}{h^2}
\]
3. **Determine \( f(x) = x^2 + 5x + 8 \) and \( h(x) = x^3 + 5x + 8 \):**
- \( f'(x) = 2x + 5 \)
- \( h'(x) = 3x^2 + 5 \)
4. **Apply the quotient rule:**
\[
u'(x) = \frac{(2x + 5)(x^3 + 5x + 8) - (x^2 + 5x + 8)(3x^2 + 5)}{(x^3 + 5x + 8)^2}
\]
5. **Plug \( u \) and \( u' \) into the derivative of the logarithmic function:**
\[
g'(x) = \frac{1}{u(x)} \cdot u'(x)
\]
Simplify this expression for the final result.
![Below is the transcription of the mathematical derivatives as shown in the image.
Equation 1:
\[ g'(x) = \frac{2x - 5}{x^2 + 5x + 8} - \frac{3x^2 - 5}{x^3 + 5x + 8} \]
Equation 2:
\[ g'(x) = \frac{x^2 + 5x + 8}{2x + 5} - \frac{x^3 + 5x + 8}{3x^2 + 5} \]
Equation 3:
\[ g'(x) = \frac{2x + 5}{x^2 + 5x + 8} + \frac{3x^2 + 5}{x^3 + 5x + 8} \]
There are no graphs or diagrams to explain in this image. The image consists solely of these mathematical expressions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4cbe10c6-7416-4dbe-aac3-86d00105e150%2F2d0f6701-7d56-4e8c-828c-dcbd2388acf5%2Ffo68cen_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Below is the transcription of the mathematical derivatives as shown in the image.
Equation 1:
\[ g'(x) = \frac{2x - 5}{x^2 + 5x + 8} - \frac{3x^2 - 5}{x^3 + 5x + 8} \]
Equation 2:
\[ g'(x) = \frac{x^2 + 5x + 8}{2x + 5} - \frac{x^3 + 5x + 8}{3x^2 + 5} \]
Equation 3:
\[ g'(x) = \frac{2x + 5}{x^2 + 5x + 8} + \frac{3x^2 + 5}{x^3 + 5x + 8} \]
There are no graphs or diagrams to explain in this image. The image consists solely of these mathematical expressions.
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