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

(a) $\frac{3 x^{2}}{2\left(x^{3}+2 y^{2}\right)}$(b) $\frac{2 y}{x^{3}+2 y^{2}}$

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 2

Partial Derivatives

Oregon State University

Harvey Mudd College

Baylor 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.

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Find the partial derivativ…

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Find the first partial der…

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Find the indicated partial…

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Find both first partial de…

so before we actually try to take the partial derivative of this, if you could simplify with some algebra, that's normally a good idea. So the first thing I'm going to do is rewrite this square root as a one half power. And then remember, if we have, like, natural log of something to a power, we can pull that power all the way out front so we can actually rewrite this as one half natural log of X cubed plus two y squared so you could take the derivative with how it was before. But we have to do change twice, and in this case, we only have to do change role once. So if you could do some algebra to simplify things, that's what I would normally suggest to do. Yes, now, though, eh? So let me just write f of x y here. So if we take the partial of this with respect to X, remember, we're going to assume that this why here is a constant. So we're going to just do the derivative like we normally would from single variable calculus. Just keep it in mind. This why here is always going to be treated as some constant. So, um, we dry dlf by Dell X But more common way to write it is f sub x two. I did the herb to indicate the partial derivative with respect tax. And then so the one half just gets pulled out front. And then remember, the derivative of natural log is gonna be one over whatever is on the inside. So one over x cubed plus two y squared. And then we have to take the partial. What's on the inside do to change rule. And now, um, the derivative of X cubed is just going to be three x squared and then the derivative of two y squared Well, why is a constant of squaring a constant so constant multiply constant So this is just zero. So we didn't write this as three x squared all over two times X cubed plus two y squared. So this here is our partial with respect to X and now to get our partial with respect to why, same thing. But we're just going to assume now that X is our Constance. Let me scoot this down. So replace that for a why and now this X here is going to be our constant so we can write del F by Dell. Why on this is gonna be equal to have subway? So again, same thing just one half over execute plus two squared. And then we would take the derivative of inside do the Channel. So Dell, by Delta y of x cubed plus two y squared. So again, remember this X we're assuming is a constant. So if I Cuba constant so constant, so derivative that zero. And then over here, two y squared we just use power rule So before Why Oh, and then this to in this four were just simplified it too. So that would give us our partial with respect to Why is going to be two y all over execute plus two y squared.

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