Use the information in the table to find h'(a) at the given value of a. h(x) = (- 2 f(x) ;a = 3 g(x) f(x) f'(x) g(x) g'(x) 7 1 9 -2 3 2 8 8 1 -1 3 -5 2 h'(a) = 2.
Use the information in the table to find h'(a) at the given value of a. h(x) = (- 2 f(x) ;a = 3 g(x) f(x) f'(x) g(x) g'(x) 7 1 9 -2 3 2 8 8 1 -1 3 -5 2 h'(a) = 2.
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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![To find \( h'(a) \) at the given value of \( a = 3 \), we use the function definition:
\[ h(x) = \left( \frac{f(x)}{g(x)} \right)^2 \]
We need to apply the chain rule and quotient rule to differentiate \( h(x) \).
**Table Information:**
The table provides values for functions and their derivatives at specific \( x \)-values:
\[
\begin{array}{|c|c|c|c|c|}
\hline
x & f(x) & f'(x) & g(x) & g'(x) \\
\hline
0 & 7 & 5 & 0 & 2 \\
1 & 9 & -2 & 3 & 0 \\
2 & 8 & 8 & 1 & -1 \\
3 & 5 & -5 & 2 & 5 \\
\hline
\end{array}
\]
**Steps to find \( h'(3) \):**
1. **Find \( h(x) \):**
\[ h(x) = \left( \frac{f(x)}{g(x)} \right)^2 \]
2. **Derivative using the chain rule and quotient rule:**
\[
h'(x) = 2 \left( \frac{f(x)}{g(x)} \right) \cdot \left( \frac{g(x)f'(x) - f(x)g'(x)}{(g(x))^2} \right)
\]
3. **Substitute \( x = 3 \) using table values:**
- \( f(3) = 5 \)
- \( f'(3) = -5 \)
- \( g(3) = 2 \)
- \( g'(3) = 5 \)
4. **Compute \( \frac{f(3)}{g(3)} \) and its derivative:**
\[
\frac{f(3)}{g(3)} = \frac{5}{2}
\]
\[
\frac{g(3)f'(3) - f(3)g'(3)}{(g(3))^2} = \frac{2(-5) - 5(5)}{](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8184eabc-db64-4ff4-8675-5dfdbf4594d7%2F8cebcb35-d2a6-4d65-b393-dc89c45d54a2%2Fmmazfk9_processed.png&w=3840&q=75)
Transcribed Image Text:To find \( h'(a) \) at the given value of \( a = 3 \), we use the function definition:
\[ h(x) = \left( \frac{f(x)}{g(x)} \right)^2 \]
We need to apply the chain rule and quotient rule to differentiate \( h(x) \).
**Table Information:**
The table provides values for functions and their derivatives at specific \( x \)-values:
\[
\begin{array}{|c|c|c|c|c|}
\hline
x & f(x) & f'(x) & g(x) & g'(x) \\
\hline
0 & 7 & 5 & 0 & 2 \\
1 & 9 & -2 & 3 & 0 \\
2 & 8 & 8 & 1 & -1 \\
3 & 5 & -5 & 2 & 5 \\
\hline
\end{array}
\]
**Steps to find \( h'(3) \):**
1. **Find \( h(x) \):**
\[ h(x) = \left( \frac{f(x)}{g(x)} \right)^2 \]
2. **Derivative using the chain rule and quotient rule:**
\[
h'(x) = 2 \left( \frac{f(x)}{g(x)} \right) \cdot \left( \frac{g(x)f'(x) - f(x)g'(x)}{(g(x))^2} \right)
\]
3. **Substitute \( x = 3 \) using table values:**
- \( f(3) = 5 \)
- \( f'(3) = -5 \)
- \( g(3) = 2 \)
- \( g'(3) = 5 \)
4. **Compute \( \frac{f(3)}{g(3)} \) and its derivative:**
\[
\frac{f(3)}{g(3)} = \frac{5}{2}
\]
\[
\frac{g(3)f'(3) - f(3)g'(3)}{(g(3))^2} = \frac{2(-5) - 5(5)}{
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