1. 9.2 Partial Derivatives Find the partial derivatives of a given function. Evaluate partial derivatives. Find the four second-order partial derivatives of a function in two variables.

2. Example 1: For find In order to find, we regard x as the variable and y and z as constants.

3. Example 1 (cont.):

4. Example 2: For Treating y as a constant Treating x 2 and x as a constants

5. Example 3: For

7. Example 4: A cellular phone company has the following production function for a certain product: where p is the number of units produced with x units of labor and y units of capital. a) Find the number of units produced with 125 units of labor and 64 units of capital. b) Find the marginal productivities. c) Evaluate the marginal productivities at x = 125 and y = 64.

8. Example 4 (cont.):

12. DEFINITION: Second-Order Partial Derivatives

13. DEFINITION (cont.):

14. Example 5: For find the four second-order partial derivatives.

15. Example 5 (cont.):