Let h(x) = f(g(x)) and p(x) = g(f(x)). Use the table below to compute the following derivatives. a. h'(4) b. p'(2) 1 4 f(x) f'(x) g(x) 2 4 - 1 - 6 2 7 3 gʻ(x) 8 8 8 3.
Let h(x) = f(g(x)) and p(x) = g(f(x)). Use the table below to compute the following derivatives. a. h'(4) b. p'(2) 1 4 f(x) f'(x) g(x) 2 4 - 1 - 6 2 7 3 gʻ(x) 8 8 8 3.
Calculus: Early Transcendentals
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ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
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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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![# Derivative Computation from Table
Given functions \( h(x) = f(g(x)) \) and \( p(x) = g(f(x)) \), use the table below to compute the following derivatives:
- \( h'(4) \)
- \( p'(2) \)
### Table Values
\[
\begin{array}{|c|c|c|c|c|}
\hline
x & 1 & 2 & 3 & 4 \\
\hline
f(x) & 2 & 3 & 1 & 4 \\
\hline
f'(x) & -1 & -8 & -2 & -6 \\
\hline
g(x) & 3 & 1 & 4 & 2 \\
\hline
g'(x) & \frac{1}{8} & \frac{7}{8} & \frac{5}{8} & \frac{3}{8} \\
\hline
\end{array}
\]
### Problem Solutions
1. **\( h'(4) \)**
For \( h(x) = f(g(x)) \), use the chain rule to find the derivative:
\[
h'(x) = f'(g(x)) \cdot g'(x)
\]
Applying this to \( x = 4 \):
- \( g(4) = 2 \)
- \( f'(g(4)) = f'(2) = -8 \)
- \( g'(4) = \frac{3}{8} \)
Therefore,
\[
h'(4) = f'(g(4)) \cdot g'(4) = (-8) \cdot \left(\frac{3}{8}\right) = -3
\]
\[
h'(4) = -3
\]
2. **\( p'(2) \)**
For \( p(x) = g(f(x)) \), use the chain rule to find the derivative:
\[
p'(x) = g'(f(x)) \cdot f'(x)
\]
Applying this to \( x = 2 \):
- \( f(2) = 3 \)
- \( g'(f(2)) = g'(3) = \frac{](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa1de489e-8c51-4ba0-8d15-cf142b4b6c4d%2Fcc1b10f1-ed84-4abf-8c7b-ec56c225bc6a%2Fj8kxfze.png&w=3840&q=75)
Transcribed Image Text:# Derivative Computation from Table
Given functions \( h(x) = f(g(x)) \) and \( p(x) = g(f(x)) \), use the table below to compute the following derivatives:
- \( h'(4) \)
- \( p'(2) \)
### Table Values
\[
\begin{array}{|c|c|c|c|c|}
\hline
x & 1 & 2 & 3 & 4 \\
\hline
f(x) & 2 & 3 & 1 & 4 \\
\hline
f'(x) & -1 & -8 & -2 & -6 \\
\hline
g(x) & 3 & 1 & 4 & 2 \\
\hline
g'(x) & \frac{1}{8} & \frac{7}{8} & \frac{5}{8} & \frac{3}{8} \\
\hline
\end{array}
\]
### Problem Solutions
1. **\( h'(4) \)**
For \( h(x) = f(g(x)) \), use the chain rule to find the derivative:
\[
h'(x) = f'(g(x)) \cdot g'(x)
\]
Applying this to \( x = 4 \):
- \( g(4) = 2 \)
- \( f'(g(4)) = f'(2) = -8 \)
- \( g'(4) = \frac{3}{8} \)
Therefore,
\[
h'(4) = f'(g(4)) \cdot g'(4) = (-8) \cdot \left(\frac{3}{8}\right) = -3
\]
\[
h'(4) = -3
\]
2. **\( p'(2) \)**
For \( p(x) = g(f(x)) \), use the chain rule to find the derivative:
\[
p'(x) = g'(f(x)) \cdot f'(x)
\]
Applying this to \( x = 2 \):
- \( f(2) = 3 \)
- \( g'(f(2)) = g'(3) = \frac{
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