Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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Rate of Change
The relation between two quantities which displays how much greater one quantity is than another is called ratio.
Slope
The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:
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Please help to calculate the partial derivatives for problem #14. Thanks 

### Calculating First-Order Partial Derivatives #### In Exercises 1–22, find ∂f/∂x and ∂f/∂y. 1. \( f(x, y) = 2x^2 - 3y - 4 \) 2. \( f(x, y) = x^2 - xy + y^2 \) 3. \( f(x, y) = (x^2 - 1)(y + 2) \) 4. \( f(x, y) = 5xy - 7x^2 - y^2 + 3x - 6y + 2 \) 5. \( f(x, y) = (xy - 1)^2 \) 6. \( f(x, y) = (2x - 3y)^3 \) 7. \( f(x, y) = \sqrt{x^2 + y^2} \) 8. \( f(x, y) = (x^3 + \frac{y}{2})^{2/3} \) 9. \( f(x, y) = \frac{1}{x + y} \) 10. \( f(x, y) = \frac{x}{x^2 + y^2} \) 11. \( f(x, y) = \frac{x + y}{xy - 1} \) 12. \( f(x, y) = \tan^{-1}(\frac{y}{x}) \) 13. \( f(x, y) = e^{x + y + 1} \) 14. \( f(x, y) = e^{-x} \sin(x + y) \) 15. \( f(x, y) = \ln(x + y) \) 16. \( f(x, y) = e^{xy} \ln y \) 17. \( f(x, y) = \sin^2(x - 3y) \) 18. \( f(x, y) = \cos^2(3x - y^2) \) In this set of exercises, you are to calculate the first-order partial derivatives of each given function \( f(x, y) \) with respect to both \( x \) and \( y \). ### Explanation Partial derivatives are used to measure how a function changes as each of its input variables changes while the other variables are held constant
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Transcribed Image Text:### Calculating First-Order Partial Derivatives #### In Exercises 1–22, find ∂f/∂x and ∂f/∂y. 1. \( f(x, y) = 2x^2 - 3y - 4 \) 2. \( f(x, y) = x^2 - xy + y^2 \) 3. \( f(x, y) = (x^2 - 1)(y + 2) \) 4. \( f(x, y) = 5xy - 7x^2 - y^2 + 3x - 6y + 2 \) 5. \( f(x, y) = (xy - 1)^2 \) 6. \( f(x, y) = (2x - 3y)^3 \) 7. \( f(x, y) = \sqrt{x^2 + y^2} \) 8. \( f(x, y) = (x^3 + \frac{y}{2})^{2/3} \) 9. \( f(x, y) = \frac{1}{x + y} \) 10. \( f(x, y) = \frac{x}{x^2 + y^2} \) 11. \( f(x, y) = \frac{x + y}{xy - 1} \) 12. \( f(x, y) = \tan^{-1}(\frac{y}{x}) \) 13. \( f(x, y) = e^{x + y + 1} \) 14. \( f(x, y) = e^{-x} \sin(x + y) \) 15. \( f(x, y) = \ln(x + y) \) 16. \( f(x, y) = e^{xy} \ln y \) 17. \( f(x, y) = \sin^2(x - 3y) \) 18. \( f(x, y) = \cos^2(3x - y^2) \) In this set of exercises, you are to calculate the first-order partial derivatives of each given function \( f(x, y) \) with respect to both \( x \) and \( y \). ### Explanation Partial derivatives are used to measure how a function changes as each of its input variables changes while the other variables are held constant
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