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

$$\begin{aligned}&\text { For } f(x, y, z)=2 x^{3} y^{2} z^{2}+2 x^{2}+2 y^{2}+3 z^{2}+2 x y+3 x z+5 y z-2 x+\\&2 y+2 z, \text { find } f_{z x y}(x, y, z) \text { and } f_{x z y}(x, y, z)\end{aligned}$$

Both are $24 x^{2} y z$

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 2

Partial Derivatives

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

Boston College

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

05:22

For $f(x, y, z)=4 x^{3} y^…

11:04

Let $f(x, y, z)=x^{2} y+x …

02:35

Let $F(x, y, z)=x^{3} y z^…

01:51

Let $f(x, y, z)=x^{2} y^{4…

01:49

01:45

01:59

Let $f(x, y, z)=x^{2} y^{3…

01:29

Find $f_{x y}, f_{y z}, f_…

03:49

Let $f(x, y, z)=x^{3} y^{5…

05:52

so before we actually start differentiating this something I want toe point out is one of the the're ums in the book that says it essentially doesn't matter the order in which we take thes partials. It should all give us the same thing regardless. So if we were to kind of analyze this knows how we take a partial put respect to X partial but respect to Z and impartial blood respect toe Why? So we can take this in pretty much any order we want because this function here is going to be smooth. And if we look at it, it is essentially just a polynomial. And normally, polynomial is have all, like the really nice, um, properties we want, especially in the sense of it being smooth. And so because of that, it doesn't matter what order we actually take this in. So, um, I'm just going thio take this with respect to Z first the next and why? But you could do X Z. Why? It will give you the exact same thing. I mean, you could even done X y z, and we'll give you the same thing as up there. Um, yeah. So let's first just go ahead and call this F So now if I do del by del Z of this So if it's just like an extra y by itself, when we take these partials is going to be zero. So, like, this here would be zero. This here is zero. So x times, Why that zero x TMZ? That's not why Times e, that's not X and why, um and then everything else we could just take the partial derivative of it. So now this first term. So actually this would be a subsidy. We assume that this is a constant. So we just take the derivative X square or Z square, so would be to Z. So multiply that together. That would give us for X cubed y squared Z and then over here, taking the derivative of three z squared that would give us six Z and then over here will be assumed this X is a constant. So just take the derivative Z, which would be one. So that would be plus three X. Um Then over here, we assume this y is a constant. So again, who just be five. Why, like the last one and then driven A to Z would just be too. Okay, Next we can go ahead and take the derivative of this with respective X. So, Dell, by Dell X So this Z is zero the y zero in the 20 Then we could just take these. So remember this Why in the sea we treat as a constant so we just take the derivative X squared to that would be three x squared. Then multiply all of that together. So we have f Z X is equal to 12 x squared y squared z and then the derivative of three X will just be three then lastly, we're going to do Del Beidle. Why of this? And so the X and Z, which is a constant So we just take the derivative of y squared which would be to why and then the derivative of three is going to be zero. So multiply all that together we get eps of uh huh or F sub z x Y is equal to 24 x squared y z. And so again, this here should be equal to if we were to rearrange it in the other way. So this is going to be equal to F sub x z y as well. So if you want you to go ahead and take this partial here and you'll see you'll get the same thing. But remember, as long as it looks like a pretty nice function, um, you should be able to do this and not have to worry about which order you're actually taking the partial derivative in.

View More Answers From This Book

Find Another Textbook

03:16

Evaluate $\int_{R} \int f(x, y) d A$ for $R$ and $f$ as given.(a) $f(x, …

01:21

Linearize the given function.$$f(x, y)=x^{4}+y^{4}+16 x y \text { near }…

01:57

A function is said to be homogeneous of degree $n$ if $f(\gamma x, \gamma y)…

02:02

A producer sells two commodities, $x$ units of one commodity and $y$ units o…

06:30

For $f(x, y, z)=2 x^{2}+2 y^{2}+3 z^{2}+2 x y+3 x z+5 y z-2 x+2 y+2 z$ find …

04:40

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

01:02

Solve the given differential equation.$$\frac{d y}{d x}=3 x^{4} y^{4}$$<…

02:11

02:15

(a) Compute $\frac{d}{d x}\left(\frac{1}{b} \ln (a+b x)\right) ;$ use this t…

03:40

For $f(x, y, z)=2 x^{2}-16 x+3 y^{2}-24 y+4 z^{2}-40 z+25,$ find the point(s…