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
![**Chain Rule Explanation**
The Chain Rule is a fundamental concept in calculus used to find the derivative of a composite function. If you have two functions, \( f(x) \) and \( g(x) \), and a composite function \( C(x) = f(g(x)) \), the Chain Rule states that the derivative \( C'(x) \) is calculated as:
\[ C'(x) = f'(g(x)) \cdot g'(x) \]
**Problem Setup and Table**
Given the function \( C(x) = f(g(x)) \), find \( C'(1) \) using the provided table:
\[
\begin{array}{|c|c|c|c|c|}
\hline
x & f(x) & g(x) & f'(x) & g'(x) \\
\hline
1 & 3 & 2 & 4 & 6 \\
\hline
2 & 1 & 8 & 5 & 7 \\
\hline
3 & 7 & 2 & 7 & 9 \\
\hline
\end{array}
\]
**Solution Steps**
1. **Identify \( g(1) \):**
- From the table, when \( x = 1 \), \( g(x) = 2 \).
2. **Find \( f'(g(1)) \):**
- Since \( g(1) = 2 \), look for \( f'(x) \) when \( g(x) = 2 \).
- From the table, \( f'(x) = 7 \) when \( x = 3 \) and \( g(x) = 2 \).
3. **Identify \( g'(1) \):**
- From the table, when \( x = 1 \), \( g'(x) = 6 \).
4. **Apply the Chain Rule:**
\[
C'(1) = f'(g(1)) \cdot g'(1) = 7 \cdot 6 = 42
\]
Therefore, using the Chain Rule and the given table, we find that \( C'(1) = 42 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6a2a18af-6188-482f-813a-e68f87e43dd9%2Fc196a0b3-0ea7-44a8-86ad-5a483f47214d%2Fd1d6xzz_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Chain Rule Explanation**
The Chain Rule is a fundamental concept in calculus used to find the derivative of a composite function. If you have two functions, \( f(x) \) and \( g(x) \), and a composite function \( C(x) = f(g(x)) \), the Chain Rule states that the derivative \( C'(x) \) is calculated as:
\[ C'(x) = f'(g(x)) \cdot g'(x) \]
**Problem Setup and Table**
Given the function \( C(x) = f(g(x)) \), find \( C'(1) \) using the provided table:
\[
\begin{array}{|c|c|c|c|c|}
\hline
x & f(x) & g(x) & f'(x) & g'(x) \\
\hline
1 & 3 & 2 & 4 & 6 \\
\hline
2 & 1 & 8 & 5 & 7 \\
\hline
3 & 7 & 2 & 7 & 9 \\
\hline
\end{array}
\]
**Solution Steps**
1. **Identify \( g(1) \):**
- From the table, when \( x = 1 \), \( g(x) = 2 \).
2. **Find \( f'(g(1)) \):**
- Since \( g(1) = 2 \), look for \( f'(x) \) when \( g(x) = 2 \).
- From the table, \( f'(x) = 7 \) when \( x = 3 \) and \( g(x) = 2 \).
3. **Identify \( g'(1) \):**
- From the table, when \( x = 1 \), \( g'(x) = 6 \).
4. **Apply the Chain Rule:**
\[
C'(1) = f'(g(1)) \cdot g'(1) = 7 \cdot 6 = 42
\]
Therefore, using the Chain Rule and the given table, we find that \( C'(1) = 42 \).
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