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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**Example Problem: Calculating the Derivative**
Use the given information to find \( f '(9) \).
\[ f(x) = 3g(x) + h(x) \]
\[ g(9) = -4 \quad \text{and} \quad g'(9) = -1 \]
\[ h(9) = -5 \quad \text{and} \quad h'(9) = -9 \]
\[ f '(9) = \]
---
**Solution:**
To find \( f '(9) \), we need to differentiate \( f(x) \).
Given:
\[ f(x) = 3g(x) + h(x) \]
Differentiate both sides with respect to \( x \):
\[ f '(x) = 3g'(x) + h'(x) \]
Now, substitute \( x = 9 \):
\[ f '(9) = 3g'(9) + h'(9) \]
Given \( g'(9) = -1 \) and \( h'(9) = -9 \):
\[ f '(9) = 3(-1) + (-9) \]
\[ f '(9) = -3 - 9 \]
\[ f '(9) = -12 \]
So,
\[ f '(9) = -12 \]
---
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Transcribed Image Text:---
**Example Problem: Calculating the Derivative**
Use the given information to find \( f '(9) \).
\[ f(x) = 3g(x) + h(x) \]
\[ g(9) = -4 \quad \text{and} \quad g'(9) = -1 \]
\[ h(9) = -5 \quad \text{and} \quad h'(9) = -9 \]
\[ f '(9) = \]
---
**Solution:**
To find \( f '(9) \), we need to differentiate \( f(x) \).
Given:
\[ f(x) = 3g(x) + h(x) \]
Differentiate both sides with respect to \( x \):
\[ f '(x) = 3g'(x) + h'(x) \]
Now, substitute \( x = 9 \):
\[ f '(9) = 3g'(9) + h'(9) \]
Given \( g'(9) = -1 \) and \( h'(9) = -9 \):
\[ f '(9) = 3(-1) + (-9) \]
\[ f '(9) = -3 - 9 \]
\[ f '(9) = -12 \]
So,
\[ f '(9) = -12 \]
---
If you need further assistance, click the "Read It" button below.
[Read It]
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