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## Partial Derivatives

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**Partial Derivatives**Written by Dr. Julia Arnold Professor of Mathematics Tidewater Community College, Norfolk Campus, Norfolk, VA With Assistance from a VCCS LearningWare Grant**In this lesson you will learn**• about partial derivatives of a function of two variables • about partial derivatives of a function of three or more variables • higher-order partial derivative**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 fx and fy defined by Provided the limits exist.**To find the partial derivatives, hold one variable constant**and differentiate with respect to the other. Example 1: Find the partial derivatives fx and fy for the function**To find the partial derivatives, hold one variable constant**and differentiate with respect to the other. Example 1: Find the partial derivatives fx and fy for the function Solution:**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**Example 2: Find the partials fx and fy and evaluate them at**the indicated point for the function**Example 2: Find the partials fx and fy and evaluate them at**the indicated point for the function Solution:**The slide which follows shows the geometric interpretation**of the partial derivative. For a fixed x, z = f(x0,y) represents the curve formed by intersecting the surface z = f(x,y) with the plane x = x0. represents the slope of this curve at the point (x0,y0,f(x0,y0)) In order to view the animation, you must have the power point in slide show mode. Thanks to http://astro.temple.edu/~dhill001/partial-demo/ For the animation.**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**Notation for Higher Order Partial Derivatives**Below are the different 2nd 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**Theorem**If f is a function of x and y such that fxy and fyx are continuous on an open disk R, then, for every (x,y) in R, fxy(x,y)= fyx(x,y) Example 3: Find all of the second partial derivatives of Work the problem first then check.**Example 3:**Find all of the second partial derivatives of Notice that fxy = fyx**Example 4: Find the following partial derivatives for the**function a. b. c. d. e. Work it out then go to the next slide.**Example 4: Find the following partial derivatives for the**function a. Again, notice that the 2nd partials fxz = fzx b.**c.**e. Notice All Are Equal d.**For comments on this presentation you may email the author**Dr. Julia Arnold at jarnold@tcc.edu.