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Find the partial derivatives with respect to (a) $x,$ (b) $y$ and (c) $z$.$$f(x, y, z)=\ln \left(2 x^{2}+3 y^{2}+4 z^{2}\right)$$

(a) $\frac{4 x}{2 x^{2}+3 y^{2}+4 z^{2}}$(b) $\frac{6 y}{2 x^{2}+3 y^{2}+4 z^{2}}$(c) $\frac{8 z}{2 x^{2}+3 y^{2}+4 z^{2}}$

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 2

Partial Derivatives

Campbell University

Harvey Mudd College

Idaho State University

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

04:04

Find the partial derivativ…

09:45

04:43

05:00

03:41

02:43

04:00

Find the indicated partial…

00:45

Find the first partial der…

06:48

So for us to take the partial derivative of this, remember, what we're going to assume is all other variables are going to be a constant. So if I want to take the partial derivative of this with respect X So we applied Del by Dell X on each side. So this will give us LF by Dell X were just f sub X. Now we would first just take the derivative like we normally would s O that would be one over what's on the inside due to the natural log. So two x squared plus three y squared plus four z square And now chain rule says we do del by Dell extra just the derivative of what's on the inside. Okay. And now what we're going to assume is that everything other than X is a constant. So this three y squared is going to end up being a constant because why square times three constant If y is a constant same thing over here, if Z is a constant, then if I square con still constant multiplied by constant so constant. So when I take these derivatives here, those are just going to be zero and then I just take the derivative to explore like we would normally using power rules. So just before X, So over here, this is going to give us one of er not one, uh, four x over to expert plus three y squared plus four z squared. And then if we want to find the partial with respect, why, it will be the same process. So we just come over here and do del by Dell. Why on each side's window with Dell F by Dell, Why hope? Actually, I've got to write that this is f sub x here. So this is our partial retrospectives. Mhm on, then back to taking the partial, introspective y so f sub y. So again we just take the driver like they normally would s would be won over to execute plus three y squared plus four c squared that we have Dell by Dell Why of two x squared plus three y squared plus four z squared. And now eggs is a constant with respect to why so if I square it, multiply it by some number Still constants of zero Same thing over here with the sward once again and then the derivative of this. Well, we just use power will sort of be six. Why? I would go ahead and write this out so it be f y or the partial derivative with respect to y is equal to six y all over two x squared plus three wise respectable plus three y squared plus four z squared into This is our partial with respect to why And then lastly for partial respect, Dizzy. Same thing we just did. But now we're going to assume that ex. And why are Constance So would you del by Del Z So in it with LF by Del Z is it too f sub z which would be one over two x squared plus three y squared plus four z squared and then del by del Z of two Expert plus three y squared plus or Z squared. And now the X and the wise we're assuming are constants. So X squared times too so constant, so derivative, that zero same thing with y squared zero. And then when we take the derivative of four z square, that would be eight z. So then we just go animals by the laptops or partial with respect Dizzy they're going to be eight z all over two X Square plus three y squared plus four z squared. And then that is our personal derivative with respect to Z. So again, all we really do is we take the derivative like we normally would in a single variable case, and we just assume all other variables are just like a constant.

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