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

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

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 2

Partial Derivatives

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University of Michigan - Ann Arbor

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.

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Find the partial derivativ…

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Find the first partial der…

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if we want to find the partial derivatives of this with respectable X and why so if we first to do it with respect to X, remember, what we're going to do is assume all of the wise are constants. With respect to X eso, we would normally write this as either DLF by Dell X or we would just write f sub x Um And then over here we would take the derivative like we normally would. So natural laws derivatives should be won over whatever is on the inside. So one over two x squared plus three y squared. But then we have to take the derivative on the inside. Do the chain rule. Eso Let me just write that up. Del by Dell ex of two X squared plus three y squared So now two x squared we take the derivative just like we did in single variable calculus. So just before X and then three y squared Well, why is it constantly perspective XO squaring that so constant multiplied by constant constant. So the derivative of that is just zero. So we end up with our partial with respect. Tax is going to be for X over two x squared plus three y squared. That's our partial with respect to X Now to take the partial with respect. Why, it'll be the same thing. But we're just going to assume now that X is our constant. We'll just treat that as if it was a constant. So get rid of that, Yes, so this X here we're assuming, is our constant. So we would write Della by del y or F sub y. And again, it'll just be one over what we have on the inside there. And then we do del by del y of what's on the inside Do the change room. And so now remember, we're assuming X is a constant now. So x squared times two is a constant overall. So that would be zero and then three y squared. That would give us six y. So then, that gives us that are partial with respect to f r R. Partial with respect to why is going to be six y over two x squared plus six y squared. And so again, remember, when we're taking these partials, all we really need to remember is that we treat all the other variables as if they are constants and just take the derivative just like we would in the single variable case

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