Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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![**Section 12.7 Higher-Order Derivatives**
### Explanation
The page is from a calculus textbook, focusing on higher-order derivatives. It includes a worked-out example, followed by practice problems.
**Worked Example**
The text demonstrates the process of finding higher-order derivatives of a given implicit function \(2y = e^{xy}\).
- Start with the original equation: \(y^3 = e^{3y}\).
- Differentiate implicitly to find \(\frac{dy}{dx}\).
- Apply the chain rule and solve for \(\frac{d^2y}{dx^2}\).
- Express the final result without \(\frac{dy}{dx}\).
Final expression:
\[
\frac{d^2y}{dx^2} = \frac{2\left(\frac{y - e^y}{2}\right)}{(2 - y)^2} \cdot \frac{2y}{2 - y}
\]
### Practice Problems 12.7
**Problems 1-20: Find the Indicated Derivatives**
1. \(y = 4x^4 - 12x^2 + (x + 2), y'\)
2. \(y = x^4 - x^3 + x^2 + x + 1, y'\)
3. \(y = x^5 + x^4 + x, y'\)
4. \(y = \sin(q + \ln q), \frac{dq}{d\theta}\)
5. \(y = x^2 + 3 \cdot \frac{d^2x}{dx^2}\)
6. \(F(g) = \ln(q + t), \frac{dF}{dg}\)
7. \(f(s) = 3 \ln t, y''\)
8. \(y = \frac{1}{x}, y''\)
9. \(f(g) = \ln \left(2q \cos q\right)\)
10. \(f(x) = \sqrt{x} \cdot f''(x)\)
11. \(f(q) = \sqrt{r} \cdot f'(r)\)
12. \(y = e^{x^2}, y''\)
13. \(y = \frac{2x + 3}{x^4}\)
14. \(y = (3x +](https://content.bartleby.com/qna-images/question/0e2d6f28-e3d7-414c-b1d4-a752131175c2/82cfab03-b4f2-4799-9d2e-b218a0ad8d2e/18i1eja.jpeg)
Transcribed Image Text:**Section 12.7 Higher-Order Derivatives**
### Explanation
The page is from a calculus textbook, focusing on higher-order derivatives. It includes a worked-out example, followed by practice problems.
**Worked Example**
The text demonstrates the process of finding higher-order derivatives of a given implicit function \(2y = e^{xy}\).
- Start with the original equation: \(y^3 = e^{3y}\).
- Differentiate implicitly to find \(\frac{dy}{dx}\).
- Apply the chain rule and solve for \(\frac{d^2y}{dx^2}\).
- Express the final result without \(\frac{dy}{dx}\).
Final expression:
\[
\frac{d^2y}{dx^2} = \frac{2\left(\frac{y - e^y}{2}\right)}{(2 - y)^2} \cdot \frac{2y}{2 - y}
\]
### Practice Problems 12.7
**Problems 1-20: Find the Indicated Derivatives**
1. \(y = 4x^4 - 12x^2 + (x + 2), y'\)
2. \(y = x^4 - x^3 + x^2 + x + 1, y'\)
3. \(y = x^5 + x^4 + x, y'\)
4. \(y = \sin(q + \ln q), \frac{dq}{d\theta}\)
5. \(y = x^2 + 3 \cdot \frac{d^2x}{dx^2}\)
6. \(F(g) = \ln(q + t), \frac{dF}{dg}\)
7. \(f(s) = 3 \ln t, y''\)
8. \(y = \frac{1}{x}, y''\)
9. \(f(g) = \ln \left(2q \cos q\right)\)
10. \(f(x) = \sqrt{x} \cdot f''(x)\)
11. \(f(q) = \sqrt{r} \cdot f'(r)\)
12. \(y = e^{x^2}, y''\)
13. \(y = \frac{2x + 3}{x^4}\)
14. \(y = (3x +
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