Partial Derivatives with Two Variables Objectives At the end of this lesson you should be able to 1 Find the first and second order partial derivatives 2 Evaluate the first and second order partial derivatives Background A function of one variable f x x 2 2 x is graphed on the right This is shown in two dimensions with x in the horizontal and y in the vertical as usual Its derivative f x 2 x 2 is a new function that defines the slope of the curve at any point in its domain The derivative represents the rate of change of y f x with respect to wrt x Notice that the crosshairs at x 1 is at the vertex of the parabola The vertical line crosses the x axis at x y 1 0 This is no coincidence since the function f x has an extreme point at the vertex Discussion Given a function z f x y in two independent variables the idea of derivative changes a little We now consider the rate of change of z wrt x while holding y constant and the rate of change of z wrt y while holding x constant This means that we now have two derivatives because we have change created in both the x and the y directions They are called partial derivatives because we only talk about part of the change going on in the function z z f y x y f x x y and These partial derivatives are written as y x Notice the use of the Greek letter delta rather than a d as in dx This indicates a partial derivative Example Suppose z x 2 y 2 5 x 8 y 4 Let s find the two partial derivatives z we say we fix y Don t worry it doesn t hurt we just treat y like a constant x z 2x 5 Recall that the derivative of a constant is zero So x z z 2y 8 Similarly for we fix x Then y y For Arizona State University Department of Mathematics and Statistics 1 Partial Derivatives with Two Variables Formal Definitions of Partial Derivatives If z f x y then the f x h y f x y h f x y k f x y The partial derivative of z wrt y is z y f y x y lim k 0 k The partial derivative of z wrt x is z x f x x y lim h 0 So what does it all mean The partial wrt x f x x y is approximately equal to the change in f x y that results from increasing x by one unit while holding y constant The partial wrt y f y x y is approximately equal to the Pay particular attention to all of the different ways of annotating the partials z z z x z y fx fy x y This is consistent with the two dimensional situations and just as confusing change in f x y that results from increasing y by one unit while holding x constant Linear Approximations with Partials We have the same sort of linear approximations as in the 2 dimensional world However we get one for each direction of change f x dx y f x y When h dx is small f x x y gives f x x y dx f x dx y f x y dx f x y dy f x y When k dy is small f y x y gives f x x y dy f x y dy f x y dy Evaluating Partials Example Given the function z x 2 y 2 5 x 8 y 4 from above we found partial derivatives of z z 2 y 8 Let s evaluate them at x y 2 1 2 x 5 and y x z 2 1 f 2 1 22 12 5 2 8 1 4 3 So the point 2 1 3 is on the surface We can find the slope of the surface in the x direction and in the y direction by using the partial derivatives Saying z 2 1 9 means that from the point 2 1 3 on the surface the slope in the x direction is 9 z x 2 1 f x 2 1 2 2 5 9 and z y 2 1 f y 2 1 2 1 8 6 x or if x increases from 2 to 3 then the function increases by 9 units from 3 to 12 Similarly z y 2 1 6 means that from the point 2 1 3 on the surface the slope in the y direction is 6 or if y increases from 1 to 2 then the function decreases by 6 units from 3 to 6 Arizona State University Department of Mathematics and Statistics 2 Partial Derivatives with Two Variables Higher Order Partial Derivatives If z f x y then the first order partial derivatives are z and z Just as in the two dimensional y x situation we can take the derivative of these derivatives The word order is usually included when talking about partial derivatives whether it is the first second third etc of derivatives The second order partial derivatives can be taken wrt x or y That leads to the following four calculations Two are pure then then x x y y z xx f xx x y f 2 f x x x 2 The other two are mixed z yy f yy x y f 2 f y y y 2 then then Note the way we indicate them with subscripts x x y y f 2 f x y x y z y x f yx x y and and z xy f xy x y f 2 f y x y x 2 f 2 f The last two are equal x y y x Example Let f x y 5 ye x Find all first and second order partials 2 The first order partial derivatives are 2 f x 5ye x 2x 10xye x f y 5e x 2 Note the applications of the chain rule 2 Note that this is the same as if we had just had y kx with k 5e x 2 The second order partial derivatives 2 2 2 2 f xx 10 ye x 10xy2xe x 10 ye x 20x 2 ye x We need to apply the product rule 2 f yy 0 f xy 10xe x 2 No y variable in f y 5e x f y 5e x is constant wrt y 2 2 2 2 Again this is because f x 10xye x 10xe x y is linear wrt y f yx 2x5e x 10xe x 2 Note the application of the chain rule 2 Also compare back to f xy 10xe x We said they would be equal Arizona State University Department of Mathematics and Statistics 3