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- 1. Chapter 8: Partial Derivatives Section 8.3 Written by Dr. Julia Arnold Associate Professor of Mathematics Tidewater Community College, Norfolk Campus, Norfolk, VA With Assistance from a VCCS LearningWare Grant
- 2. <ul><li>In this lesson you will learn </li></ul><ul><li>about partial derivatives of a function of two variables </li></ul><ul><li>about partial derivatives of a function of three or more variables </li></ul><ul><li>higher-order partial derivative </li></ul>
- 3. Partial derivatives are defined as derivatives of a function of multiple variables when all but the variable of interest are held fixed during the differentiation. Definition of Partial Derivatives of a Function of Two Variables If z = f(x,y), the the first partial derivatives of f with respect to x and y are the functions f x and f y defined by Provided the limits exist.
- 4. To find the partial derivatives, hold one variable constant and differentiate with respect to the other. Example 1: Find the partial derivatives f x and f y for the function
- 5. To find the partial derivatives, hold one variable constant and differentiate with respect to the other. Example 1: Find the partial derivatives f x and f y for the function Solution:
- 6. Notation for First Partial Derivative For z = f(x,y), the partial derivatives fx and fy are denoted by The first partials evaluated at the point (a,b) are denoted by
- 7. Example 2: Find the partials f x and f y and evaluate them at the indicated point for the function
- 8. Example 2: Find the partials f x and f y and evaluate them at the indicated point for the function Solution:
- 9. The following slide shows the geometric interpretation of the partial derivative. For a fixed x, z = f(x 0 ,y) represents the curve formed by intersecting the surface z = f(x,y) with the plane x = x 0. represents the slope of this curve at the point (x 0 ,y 0 ,f(x 0 ,y 0 )) Thanks to http://astro.temple.edu/~dhill001/partial-demo/ For the animation.
- 11. Definition of Partial Derivatives of a Function of Three or More Variables If w = f(x,y,z), then there are three partial derivatives each of which is formed by holding two of the variables In general, if where all but the kth variable is held constant
- 12. Notation for Higher Order Partial Derivatives Below are the different 2 nd order partial derivatives: Differentiate twice with respect to x Differentiate twice with respect to y Differentiate first with respect to x and then with respect to y Differentiate first with respect to y and then with respect to x
- 13. Theorem If f is a function of x and y such that f xy and f yx are continuous on an open disk R, then, for every (x,y) in R, f xy (x,y)= f yx (x,y) Example 3: Find all of the second partial derivatives of Work the problem first then check.
- 14. Example 3: Find all of the second partial derivatives of Notice that f xy = f yx
- 15. Example 4: Find the following partial derivatives for the function a. b. c. d. e. Work it out then go to the next slide.
- 16. Example 4: Find the following partial derivatives for the function a. b. Again, notice that the 2 nd partials f xz = f zx
- 17. c. d. e. Notice All Are Equal
- 18. Go to BB for your exercises.

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