Use the given information to find f '(9). f(x) = 3g(x) + h(x) g(9) = -4 and g'(9) = -1 h(9) = -5 and h'(9) = -9 f'(9) = =

Calculus: Early Transcendentals
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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 \( f '(9) \).

\[ f(x) = 3g(x) + h(x) \]

\[ g(9) = -4 \quad \text{and} \quad g'(9) = -1 \]

\[ h(9) = -5 \quad \text{and} \quad h'(9) = -9 \]

\[ f '(9) = \]

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**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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