1. Chapter 8 Multivariable Calculus Section 2 Partial Derivatives

2. 2 Learning Objectives for Section 8.2 Partial Derivatives The student will be able to evaluate partial derivatives and second- order partial derivatives.

3. 3 Introduction to Partial Derivatives We have studied extensively differentiation of functions of one variable. For instance, if C(x) = 100 + 500x, then C ´ (x) = 500. If we have more than one independent variable, we can still differentiate the function if we consider one of the variables independent and the others fixed. This is called a partial derivative.

4. 4 Example For a company producing only one type of surfboard, the cost function is C(x) = 500 + 70x, where x is the number of boards produced. Differentiating with respect to x, we obtain the marginal cost function C ´ (x) = 70. Since the marginal cost is constant, $70 is the change in cost for a one-unit increase in production at any output level.

5. 5 Example (continued) For a company producing two types of boards, a standard model and a competition model, the cost function is C(x, y) = 700 + 70x + 100y, where x is the number of standard boards, and y is the number of competition boards produced. Now suppose that we differentiate with respect to x, holding y fixed, and denote this by C x (x, y); or suppose we differentiate with respect to y, holding x fixed, and denote this by C y (x, y). Differentiating in this way, we obtain C x (x, y) = 70, and C y (x, y)= 100.

6. 6 Example (continued) Each of these is called a partial derivative, and in this example, each represents marginal cost. The first is the change in cost due to a one-unit increase in production of the standard board with the production of the competition board held fixed. The second is the change in cost due to a one-unit increase in production of the competition board with the production of the standard board held fixed.

7. 7 Partial Derivatives If z = f (x, y), then the partial derivative of f with respect to x is defined by and is denoted by The partial derivative of f with respect to y is defined by and is denoted by

8. 8 Example Let f (x, y) = 3x 2 + 2xy – y 3. a.Find f x (x, y). [This is the derivative with respect to x, so consider y as a constant.]

9. 9 Example Let f (x, y) = 3x 2 + 2xy – y 3. a.Find f x (x, y). [This is the derivative with respect to x, so consider y as a constant.] f x (x, y) = 6x + 2y b. Find f y (x, y). [This is the derivative with respect to y, so consider x as a constant.]

10. 10 Example Let f (x, y) = 3x 2 + 2xy – y 3. a.Find f x (x, y). [This is the derivative with respect to x, so consider y as a constant.] f x (x, y) = 6x + 2y b. Find f y (x, y). [This is the derivative with respect to y, so consider x as a constant.] f y (x, y) = 2x – 3y 2 c. Find f x (2,5).

11. 11 Example Let f (x, y) = 3x 2 + 2xy – y 3. a.Find f x (x, y). [This is the derivative with respect to x, so consider y as a constant.] f x (x, y) = 6x + 2y b. Find f y (x, y). [This is the derivative with respect to y, so consider x as a constant.] f y (x, y) = 2x - 3y 2 c. Find f x (2,5). f x (2,5) = 6 · 2 + 2 · 5 = 22.

12. 12 Example Using the Chain Rule Let f (x, y) = (5 + 2xy 2 ) 3. Hint: Think of the problem as z = u 3 and u = 5 + 2xy 2. a. Find f x (x, y)

13. 13 Example Using the Chain Rule Let f (x, y) = (5 + 2xy 2 ) 3. Hint: Think of the problem as z = u 3 and u = 5 + 2xy 2. a. Find f x (x, y) f x (x, y) = 3 (5 + 2xy 2 ) 2 · 2y 2 b. Find f y (x, y) The chain.

14. 14 Example Using the Chain Rule Let f (x, y) = (5 + 2xy 2 ) 3. Hint: Think of the problem as z = u 3 and u = 5 + 2xy 2. a. Find f x (x, y) f x (x, y) = 3 (5 + 2xy 2 ) 2 · 2y 2 b. Find f y (x, y) f y (x, y)) = 3 (5 + 2xy2) 2 · 4xy The chain.

15. 15 Second-Order Partial Derivatives Taking a second-order partial derivative means taking a partial derivative of the first partial derivative. If z = f (x, y), then

16. 16 Example Let f (x, y) = x 3 y 3 + x + y 2. a. Find f x (x, y).

17. 17 Example Let f (x, y) = x 3 y 3 + x + y 2. a. Find f xx (x, y). f x (x, y) = 3x 2 y 3 + 1 f xx (x, y) = 6xy 3 b.Find f xy (x, y).

18. 18 Example Let f (x, y) = x 3 y 3 + x + y 2. a. Find f x (x, y). f x (x, y) = 3x 2 y 3 + 1 f xx (x, y) = 6xy 3 b.Find f xy (x, y). f x (x, y) = 3x 2 y 3 + 1 f xy (x, y) = 9x 2 y 2

19. 19 Summary ■ If z = f (x, y), then the partial derivative of f with respect to x is defined by ■ If z = f (x, y), then the partial derivative of f with respect to y is defined by ■ We learned how to take second partial derivatives.