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 the partial derivatives with respect to (a) $x$ and (b) $y$.$$f(x, y)=4 x^{3} y^{2}\left(5 x^{4}-2 y^{3}\right)^{6}$$

(a) $12 x^{2} y^{2}\left(5 x^{4}-2 y^{3}\right)^{5}\left(45 x^{4}-2 y^{3}\right)$(b) $40 x^{3} y\left(5 x^{4}-2 y^{3}\right)^{5}\left(x^{4}-4 y^{3}\right)$

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 2

Partial Derivatives

Missouri State University

Baylor University

University of Michigan - Ann Arbor

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

05:49

Find the partial derivativ…

05:27

05:23

03:41

06:48

01:54

Find the indicated partial…

03:21

01:01

Find all the second partia…

00:43

Find the first partial der…

05:17

05:31

05:03

02:07

Find the four second parti…

02:43

remember, in order for us to find the partial derivative of, um, some function, what we do is we assume all of the variables are going to be constant, except for the one we're trying to find the partial derivative. So if I come up here and try to take the partial of this with respect tax So I applied Del by Dell X on each side, These wise here, I'm going to treat as if they're constants. So we would write dlf by Dell X. Um, but I would say the more common notation is just f sub x like this. And then we could pull this foreign This y squares we have four y squared del by Dell X of X cubed, uh, Times five, X 2/4 minus three. Why cute about six power. So now we have a function that depends on X multiplied by function. That depends on X. So we use products room so four y squared and then times so execute. Actually, let me write this one first So five x to the fourth minus two y cute raised sixth in Del by Dell X of execute and then plus other way around so exc you'd del by Dell, ex of five X to the fourth finest to, like, cute race. The sixth eso this first derivative here would just be like power room. So three x squared. And then over here, we're gonna have to use chain rule. But when we use chain rule, we have to remember that this why here is going to be a constant. So first, we have six, um, five backs to the fourth minus two, y que grace the fifth. And then when we take the derivative of the inside here, so we would do power rule for the first part. So be 20 x cubed. But then remember, since why is a constant, um, constant cute multiplied by some concepts. So concepts that just b plus zero. Okay, now, let's go ahead and factors some of this stuff out just to kind of clean things up a little bit. Um, and I'll just factor out these powers of five here, Um, and then I'll kind of leave everything else after that. So we have four. Why, Cube? There'll be times five x to the fourth minus two y cube race. And then if we do that we'll be left with one of those in three x squared. So we have three X squared times five x to the fourth. Finest to why cubed. And then over here, if we factor out all five of those we would be left with so 20 times six times x cubed. So that would be plus 1 20 x cute on Ben. It looks like we can factor out a three and an X squared from both of these. So that would be 12 x squared y squared times five x to the fourth minus two y Cube race the fifth. And then times, um, I would just be five x to the fourth minus two y cubed. And then over here, if we factor out that three, that would be plus 40 x cute as though that this here is going to be our partial derivative that's actually are a are partial derivative with respect to X. And then let me come up here, scoop this down. So now when we want to take the partial, this with respect to why, instead of assuming these wiser constant, we're going to assume the exes are going to be constant instead. This time Oh, actually, I think there is one thing I gotta messed up up here earlier. This here shouldn't be ex cute because we have an x cubed here and here. So this should actually be X to the six eso when we factor out that x square. That should be X toothy. Fourth here so that we can combine those. Um, yeah. So I should let me go ahead and rewrite this really quickly. Sorry about that. But I just kind of caught that so we can add those together. So it be 12 x squared. Why? Square five x to the fourth minus two y cubed to the fifth and then 45 x to the fourth minus two. Y cute. Yeah, I'd say the hardest part of this is just keeping track of all of your things floating around. Yeah. So that actually will be the final answer for that. Um, and everything else looks fine. Other than that. Okay, But what I was saying down here, though, to take the partial respect Why we assume those X is there going to be constant so we could go ahead and do the same? First step that we did before kind of pull out that four x cubed eso. This will be del f of del y is equal to just f sub y, then four x cubed and Dell by Dell. Why of y squared times five x to the fourth minus two y cubed race and six and again we have two functions multiplied together. So we will need to use product rule so we would have five x to the fourth minus about three to why cute raised to six del by del y of y squared. And then plus of why squared del Beidle y of five x 2/4 minus two Cute race and six now taking the derivative of Why square that should give us to Why e even over here to take this derivative. So we use general just like we did before. So would be six times five x to the fourth minus two like you raise the fifth and then when we come over here to take the driver of the inside Now we assume this X is a constant, so that would be zero minus six y squared eso Let's go ahead and pull out that five x to the fourth, minus three White to the fifth. Like we did before and doing that, we should get some four x cubed times five x to the fourth finest to, like Cube raised to the fifth. Um, And then this first term we would just have six y times five x to the fourth minus two y cute. And then over here all those would go away. Then we need to multiply everything else together. So it would be negative. 36. And then why square times y squared? So why to the fourth? Um And then when you go ahead and factor out a to Y from each of these so that would give us eight Execute why five backs to the fourth minus two y cube race to the fifth on then five x to the fourth minus two cubed. And we factor out of two wise that would now be minus 18 y to the fourth. And then we could go ahead and combine those together. So you get eight x cube by times five x to the book, not to the fourth minus two y cube race the fifth and then negative 13 Why? To the fourth minus two. Why cute? Um, I guess you could factor out of negative if you want. Um, but I'll just go ahead and leave it like that. There are actually I added the wrong ones here. Um, let me do that again. Because I'm not adding the five X and forth amounting that negative two y cube and then a factor out the why? For whatever reason, from here. Yeah, as a five x to the fourth minus 20 y Q b A. I don't know why I'm having such a hard time with the algebra today. Um, yeah. So you could go ahead and factor out of five from this if you want, Um, so I'll just do that as well. So it be 40 x cubed. Why? Times five x to the fourth minus two. Like cube, raise the fifth, and then next to the fourth minus for Why, cute. Then this will be our partial with respect to why, um so, yeah, the majority of this troubles with this aren't necessarily taking the derivative. Is just being careful with your algebra? Because you could take is I just wanted to drop random things at certain points. Yeah, so those would be our partials

View More Answers From This Book

Find Another Textbook

00:58

$f(x, y)=x^{2}-y^{2}-3 x^{3} y,$ determine (a) $f(1,-1),(\mathrm{b}) f(2,3)$…

03:10

Find (a) $f_{x}(x, y),$ (b) $f_{y}(x, y)$ at the indicated point.$$f(x, …

02:40

Evaluate the given integral.$$\int \frac{4 x^{2}-x+4}{x^{2}+1} d x$$

02:32

Determine the derivative.$$f(x)=x^{2} e^{2 x}-2 x e^{2 x}+4 e^{2 x}$$

03:43

Differentiate implicitly and then determine the equation of the tangent line…

01:25

Evaluate the given integral.$$\int(2 x+1)^{9} 2 d x$$

00:59

$f(x, y)=100 x^{1 / 4} y^{3 / 4},$ determine (a) $f(1,16),(\text { b) } f(16…

01:35

Use the properties about odd and even functions to evaluate the given integr…

00:52

Find the critical points.$$f(x, y, z)=2 x^{2}-16 x+3 y^{2}-24 y+4 z^{2}-…

02:18

Find and classify, using the second partial derivative test, the critical po…