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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Question
![**Problem:**
b. \( f(x) = \ln \left( x - \frac{x^3}{6} + 15 \right) \)
**Question:** Find \( f'(x) \).
---
**Solution:**
Apply the chain rule to find the derivative \( f'(x) \).
1. **Identify the inner function, \( u(x) \):**
\( u(x) = x - \frac{x^3}{6} + 15 \)
2. **Differentiate \( u(x) \):**
\[
u'(x) = 1 - \frac{3x^2}{6} = 1 - \frac{x^2}{2}
\]
3. **Use the chain rule:**
The derivative of \( \ln(u) \) is \( \frac{1}{u} \times u'(x) \).
4. **Calculate \( f'(x) \):**
\[
f'(x) = \frac{1}{x - \frac{x^3}{6} + 15} \times \left( 1 - \frac{x^2}{2} \right)
\]
The result for the derivative is:
\[
f'(x) = \frac{1 - \frac{x^2}{2}}{x - \frac{x^3}{6} + 15}
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4d09ffa2-b6c2-4e97-915d-74d176d987ab%2Fc42ba092-b780-43b3-a80e-07784706ad3b%2Ftvhw8ie_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem:**
b. \( f(x) = \ln \left( x - \frac{x^3}{6} + 15 \right) \)
**Question:** Find \( f'(x) \).
---
**Solution:**
Apply the chain rule to find the derivative \( f'(x) \).
1. **Identify the inner function, \( u(x) \):**
\( u(x) = x - \frac{x^3}{6} + 15 \)
2. **Differentiate \( u(x) \):**
\[
u'(x) = 1 - \frac{3x^2}{6} = 1 - \frac{x^2}{2}
\]
3. **Use the chain rule:**
The derivative of \( \ln(u) \) is \( \frac{1}{u} \times u'(x) \).
4. **Calculate \( f'(x) \):**
\[
f'(x) = \frac{1}{x - \frac{x^3}{6} + 15} \times \left( 1 - \frac{x^2}{2} \right)
\]
The result for the derivative is:
\[
f'(x) = \frac{1 - \frac{x^2}{2}}{x - \frac{x^3}{6} + 15}
\]
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