Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Get the answer to your homework problem.

Try Numerade free for 7 days

Like

Report

Find (a) $f_{x x}(x, y),$ (b) $f_{y y}(x, y),$ (c) $f_{x y}(x, y),$ and $f_{y x}(x, y)$ for the function defined in the given Exercises.Exercise 3

(a) $18 x-48 x^{2}+30 x y^{5}$(b) $100 x^{3} y^{3}+18$(c) $-75 x^{2} y^{4}$

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 2

Partial Derivatives

Harvey Mudd College

Baylor University

University of Nottingham

Idaho State University

Lectures

12:15

In calculus, partial derivatives are derivatives of a function with respect to one or more of its arguments, where the other arguments are treated as constants. Partial derivatives contrast with total derivatives, which are derivatives of the total function with respect to all of its arguments.

04:14

06:28

Find (a) $f_{x x}(x, y),$ …

03:58

04:40

13:36

11:40

13:40

04:25

05:11

For the following exercise…

01:00

These exercises are concer…

03:11

Find $f_{x x}(x, y), f_{x …

01:50

03:03

01:53

These exercises involve fu…

01:26

So if you haven't done exercise three or at least watched video on it, I would go do that first, because otherwise, um, starting from these partials here might be a little weird from where I got these. Um, but I'm just going to kind of use the work. We've already done their kind of bootstrap ourselves, right? So if we want to first find the second full partial derivative of this will be just apply del by Dell X. I'm excited Here, give us our second partial and then remember, we're just going to assume like this. Why here is a constant and then everything else We just take the derivative like we normally would like. We'll find the second derivative in the single variable case. So it be 18 x minus 48 x squared. And then this. Why remember, since we're multiplying, it's a constant. So we just take the derivative experts that would just be plus 30 x wide. And then derivatives have it. It's just going to be zero. So that is our second full partial with respect tax. Now over here to get the second full partial with respect toe. Why? We just assume this X here is going to be a constant. So that would just be 25 x cubed. And then we take the derivative of why to the force. It would be why cubed and then Times Force. Actually, I would be 100 out here. There would just be plus 18. And then this is our second or partial with respect to why now to find our mixed partial. It doesn't really matter if we go from the partial perspective X or the partial respect. Why, at least in this case, um, so we could take either of these ball just kind of do both to kind of show how we should be able to take either and get the same answer. There are some cases where this doesn't hold. Um, but I would say for the most part, we really won't have to worry about those cases. So let's just get rid of these and then just flip them. So we have tell by 0.0 x dope. I don't Why? So over here this would give us F x. Why, um and then remember all these exes here? Actually, maybe I should do this in green, but to make it kind of stand out from what we just did. So Dell Beidle, why I'll make the screens. Well, so these exes here, we assume, are all Constance. So that would just be a zero plus zero plus. And then we would do Why are the derivative y to the fifth times 15. So I believe that is 75 x squared y to the fourth. And then derivative of seven is just zero right? Eso we get our mixed partial Bigg 75 times x squared y to the fourth. And as long as we did this right over here, if we take the partial with respect to X now, we should get the same thing. Um, so remember these wise we're going to assume or cost us now, so that would give us f Y X is equal to. So we used the power rules. That would be 75 x squared. Why? To the fourth. And then that would just be zero. And so you can see how it didn't matter in which way we did this. We ended up with the same, um, mixed, partial. And so in this case, if I were actually trying to find the mixed partial, I would have just took it for Why here? Because I don't know. That just looks little bit simpler to me. But again, it doesn't really matter which one you use, because you could get the same answer regardless.

View More Answers From This Book

Find Another Textbook

04:06

Sketch some of the level curves (contours) or the surface defined by $f$.

01:54

$$\begin{aligned}&\text { If } f(x, y, z)=4 x^{2}+2 y^{3}+5 z^{5}+3 …

(a) determine a definite integral that will determine the area of the region…

02:29

Evaluate the given definite integral.$$\int_{0}^{\sqrt{15}} \frac{x}{\le…

06:19

Determine the slope of the tangent line to the surface defined by $f(x, y)=2…

Find (a) $f_{x x}(x, y),$ (b) $f_{y y}(x, y),$ (c) $f_{x y}(x, y),$ and $f_{…

04:42

Evaluate the given integral.$$\int_{0}^{1} \int_{0}^{2}\left(x^{2}+y^{2}…

01:15

Evaluate the given definite integral.(a) $\int_{a}^{b} \frac{1}{x} d x$<…

02:08

Sketch the area represented by the given definite integral.$$\int_{-1}^{…

01:17

Determine the region $R$ determined by the given double integral.$$\int_…