Directional Derivatives and Gradients 13.6

The first two slides contain the definitions and theorems that we need from 13.6 to work the exercises that follow.

The focus of this lesson is the directional derivative The focus of this lesson is the directional derivative. We use this derivative to find the slope at a point on a surface in a specified direction. Check out the graphics on the first page of 13.6 in the text. Through the gradient, we can use partial derivatives to find the directional derivative.

Example 1. Find the directional derivative of at P(-1, 5) in the direction of Solution: Find Note in this example that

Example 2. Find the directional derivative of at P(-5,-2) in the direction of Solution. First find the unit vector in the direction of v. Since

Example 3. Find the directional derivative of at P(1,1,1) in the direction of Solution.

Example 4. Find the directional derivative of in the direction of Solution.

The following links will take you to presentations on Directional Derivatives and Gradients done at other colleges. http://math.etsu.edu/Multicalc/Chap2/Chap2-7/index.htm (Directional Derivatives are covered in the Part 2 tab.) Oregon State http://mathworld.wolfram.com/DirectionalDerivative.html (This site is created by the people who make Mathematica software.) http://ocw.mit.edu/ans7870/18/18.013a/textbook/HTML/tools/tools19.html (This is done at MIT. You HAVE to check this out just for the graphics. Click on the link to get related text.)