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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Question
![**Problem Statement**
Find \( f'(9) \) if \( f(x) = \frac{7x^2 + 6x + 2}{\sqrt{x}} \).
**Task**
Calculate the derivative of the function \( f(x) \) and evaluate it at \( x = 9 \).
**Steps to Solve**
1. **Rewrite the Function:**
Express the function in terms of exponents to simplify differentiation:
\[
f(x) = \frac{7x^2 + 6x + 2}{x^{1/2}} = (7x^2 + 6x + 2) \cdot x^{-1/2}
\]
2. **Use the Product Rule:**
Recall the product rule: if \( u(x) \) and \( v(x) \) are functions, then
\[
(uv)' = u'v + uv'
\]
Assign:
\[
u(x) = 7x^2 + 6x + 2, \quad v(x) = x^{-1/2}
\]
Differentiate:
\[
u'(x) = 14x + 6, \quad v'(x) = -\frac{1}{2}x^{-3/2}
\]
3. **Apply the Product Rule:**
\[
f'(x) = u'v + uv' = (14x + 6)x^{-1/2} + (7x^2 + 6x + 2)\left(-\frac{1}{2}x^{-3/2}\right)
\]
4. **Simplify \( f'(x) \):**
Calculate each term and simplify the expression:
\[
f'(x) = (14x + 6)x^{-1/2} - \frac{1}{2}(7x^2 + 6x + 2)x^{-3/2}
\]
5. **Evaluate \( f'(9) \):**
Substitute \( x = 9 \) into the derivative function to find \( f'(9) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7ca42388-a0e6-4236-bc5e-61764756f4fd%2F88c10ef9-d8d2-4848-adc7-5783de725a57%2Fg86en4n_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement**
Find \( f'(9) \) if \( f(x) = \frac{7x^2 + 6x + 2}{\sqrt{x}} \).
**Task**
Calculate the derivative of the function \( f(x) \) and evaluate it at \( x = 9 \).
**Steps to Solve**
1. **Rewrite the Function:**
Express the function in terms of exponents to simplify differentiation:
\[
f(x) = \frac{7x^2 + 6x + 2}{x^{1/2}} = (7x^2 + 6x + 2) \cdot x^{-1/2}
\]
2. **Use the Product Rule:**
Recall the product rule: if \( u(x) \) and \( v(x) \) are functions, then
\[
(uv)' = u'v + uv'
\]
Assign:
\[
u(x) = 7x^2 + 6x + 2, \quad v(x) = x^{-1/2}
\]
Differentiate:
\[
u'(x) = 14x + 6, \quad v'(x) = -\frac{1}{2}x^{-3/2}
\]
3. **Apply the Product Rule:**
\[
f'(x) = u'v + uv' = (14x + 6)x^{-1/2} + (7x^2 + 6x + 2)\left(-\frac{1}{2}x^{-3/2}\right)
\]
4. **Simplify \( f'(x) \):**
Calculate each term and simplify the expression:
\[
f'(x) = (14x + 6)x^{-1/2} - \frac{1}{2}(7x^2 + 6x + 2)x^{-3/2}
\]
5. **Evaluate \( f'(9) \):**
Substitute \( x = 9 \) into the derivative function to find \( f'(9) \).
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for any real n
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