1. Chapter 14 – Partial Derivatives 14.3 Partial Derivatives 1 Objectives:  Understand the various aspects of partial derivatives Dr. Erickson

2. Partial Derivative w.r.t. x at (a, b) In general, if f is a function of two variables x and y, suppose we let only x vary while keeping y fixed, say y = b, where b is a constant. Then, we are really considering a function of a single variable x g(x) = f(x, b) 14.3 Partial Derivatives2Dr. Erickson

3. Partial Derivative w.r.t. x at (a, b) If g has a derivative at a, we call it the partial derivative of f with respect to x at (a, b). We denote it by: f x (a, b) 14.3 Partial Derivatives3Dr. Erickson

4. Partial Derivative w.r.t. x at (a, b) So we have, By using the definition of derivative, this equation becomes 14.3 Partial Derivatives4Dr. Erickson

5. Partial Derivative w.r.t. y at (a, b) Similarly, the partial derivative of f with respect to y at (a, b), denoted by f y (a, b), is obtained by: ◦ Keeping x fixed (x = a) ◦ Finding the ordinary derivative at b of the function G(y) = f(a, y) 14.3 Partial Derivatives5Dr. Erickson

6. Partial Derivative w.r.t. y at (a, b) So we have, 14.3 Partial Derivatives6Dr. Erickson

7. Definition - Partial Derivatives If we now let the point (a, b) vary in Equations 2 and 3, f x and f y become functions of two variables. 14.3 Partial Derivatives7Dr. Erickson

8. Notation for Partial Derivatives If z = f (x,y), we can write 14.3 Partial Derivatives8Dr. Erickson

9. Rule for finding Partial Derivatives z = f (x,y) To find f x, regard y as a constant and differentiate f (x,y) w.r.t. x. To find f y, regard x as a constant and differentiate f (x,y) w.r.t. y. 14.3 Partial Derivatives9Dr. Erickson

10. Example 1 – pg. 912 # 16 Find the first partial derivatives of the function. 14.3 Partial Derivatives10Dr. Erickson

11. Example 2 Find the first partial derivatives of the function. 14.3 Partial Derivatives11Dr. Erickson

12. Function of more than Two Variables A function of three variables has the partial derivative w.r.t. x is defined as and is found by treating y and z as constants and differentiating the function w.r.t. x 14.3 Partial Derivatives12Dr. Erickson

13. Example 3 Find the first partial derivatives of the function. 14.3 Partial Derivatives13Dr. Erickson

14. Example 4 Find the first partial derivatives of the function. 14.3 Partial Derivatives14Dr. Erickson

15. Higher Derivatives If f is a function of two variables, then its partial derivatives f x and f y are also functions of two variables. So, we can consider their partial derivatives (f x ) x, (f x ) y, (f y ) x, (f y ) y These are called the second partial derivatives of f. 14.3 Partial Derivatives15Dr. Erickson

16. Notation 14.3 Partial Derivatives16Dr. Erickson

17. Example 5 Use implicit differentiation to find  z/  x and  z/  y. 14.3 Partial Derivatives17Dr. Erickson

18. Example 6 – pg. 913 # 54 Find all the second partial derivatives. 14.3 Partial Derivatives18Dr. Erickson

19. Example 7 Find the indicated partial derivative. 14.3 Partial Derivatives19Dr. Erickson

20. Clairaut’s Theorem 14.3 Partial Derivatives20Dr. Erickson

21. Example 8 – pg. 913 # 70 Find the indicated partial derivative. 14.3 Partial Derivatives21Dr. Erickson

22. More Examples The video examples below are from section 14.3 in your textbook. Please watch them on your own time for extra instruction. Each video is about 2 minutes in length. ◦ Example 3 Example 3 ◦ Example 4 Example 4 ◦ Example 7 Example 7 14.3 Partial Derivatives22Dr. Erickson

23. Demonstrations Feel free to explore these demonstrations below. Partial Derivatives in 3D Laplace's Equation on a Circle Laplace's Equation on a Square 14.3 Partial Derivatives23Dr. Erickson