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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![### Calculating the Second and Third Derivatives
Given the function:
\[ y = 7x^4 - 7x^2 + 9x \]
**Calculate the second and third derivatives:**
1. **First Derivative (\(y'\)):**
2. **Second Derivative (\(y'')):**
\[
y'' = \boxed{}
\]
3. **Third Derivative (\(y'''))**
\[
y''' = \boxed{}
\]
**Steps to Calculate the Derivatives:**
1. **First Derivative \(y'\):**
Apply the power rule to each term to find the first derivative.
\[ y' = \frac{d}{dx}(7x^4) - \frac{d}{dx}(7x^2) + \frac{d}{dx}(9x) \]
Simplifying this, you get
\[ y' = 28x^3 - 14x + 9 \]
2. **Second Derivative \(y''\):**
Now, differentiate \( y' \) again to find the second derivative.
\[ y'' = \frac{d}{dx}(28x^3) - \frac{d}{dx}(14x) + \frac{d}{dx}(9) \]
Simplifying this, you get
\[ y'' = 84x^2 - 14 \]
3. **Third Derivative \(y'''\):**
Differentiate \( y'' \) again to find the third derivative.
\[ y''' = \frac{d}{dx}(84x^2) - \frac{d}{dx}(14) \]
Simplifying this, you get
\[ y''' = 168x \]
Fill in the boxes with these results:
- \( y'' = 84x^2 - 14 \)
- \( y''' = 168x \)
**Note:** Ensure to follow the steps mentioned above in your calculations to verify the results.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3972ee88-18a6-4abd-967b-f522c80949cc%2F3ea13db4-d993-4f64-9253-2a08946ecf09%2Feb3vrd_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Calculating the Second and Third Derivatives
Given the function:
\[ y = 7x^4 - 7x^2 + 9x \]
**Calculate the second and third derivatives:**
1. **First Derivative (\(y'\)):**
2. **Second Derivative (\(y'')):**
\[
y'' = \boxed{}
\]
3. **Third Derivative (\(y'''))**
\[
y''' = \boxed{}
\]
**Steps to Calculate the Derivatives:**
1. **First Derivative \(y'\):**
Apply the power rule to each term to find the first derivative.
\[ y' = \frac{d}{dx}(7x^4) - \frac{d}{dx}(7x^2) + \frac{d}{dx}(9x) \]
Simplifying this, you get
\[ y' = 28x^3 - 14x + 9 \]
2. **Second Derivative \(y''\):**
Now, differentiate \( y' \) again to find the second derivative.
\[ y'' = \frac{d}{dx}(28x^3) - \frac{d}{dx}(14x) + \frac{d}{dx}(9) \]
Simplifying this, you get
\[ y'' = 84x^2 - 14 \]
3. **Third Derivative \(y'''\):**
Differentiate \( y'' \) again to find the third derivative.
\[ y''' = \frac{d}{dx}(84x^2) - \frac{d}{dx}(14) \]
Simplifying this, you get
\[ y''' = 168x \]
Fill in the boxes with these results:
- \( y'' = 84x^2 - 14 \)
- \( y''' = 168x \)
**Note:** Ensure to follow the steps mentioned above in your calculations to verify the results.
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