Chapter 14 – Partial Derivatives

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

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 Derivatives

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

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 Derivatives

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

6. Partial Derivative w.r.t. yat (a, b) • So we have, 14.3 Partial Derivatives

7. Definition - Partial Derivatives • If we now let the point (a, b) vary in Equations 2 and 3, fx and fy become functions of two variables. 14.3 Partial Derivatives

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

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

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

11. Example 2 • Find the first partial derivatives of the function. 14.3 Partial Derivatives

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 Derivatives

13. Example 3 • Find the first partial derivatives of the function. 14.3 Partial Derivatives

14. Example 4 • Find the first partial derivatives of the function. 14.3 Partial Derivatives

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

16. Notation 14.3 Partial Derivatives

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

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

19. Example 7 • Find the indicated partial derivative. 14.3 Partial Derivatives

20. Clairaut’s Theorem 14.3 Partial Derivatives

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

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 4 • Example 7 14.3 Partial Derivatives

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 Derivatives