Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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**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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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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