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

(a) $3 x^{2}\left(x^{2}+y^{2}\right)^{3}\left(11 x^{2}+3 y^{2}\right)$(b) $24 x^{3} y\left(x^{2}+y^{2}\right)^{3}$

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 2

Partial Derivatives

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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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06:48

Find the partial derivativ…

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05:27

05:17

09:53

02:43

05:31

05:03

00:43

Find the first partial der…

05:21

01:31

Find the four second parti…

02:07

03:41

So in order for us to take this partial derivative here with respect to X to start. Remember, what we're going to assume is that when we work this out or Dell by Delta X of this this Why here? We assume it is a constant. So everything else is a like actual variable. But we're just going to assume that this why is a constant So any kind of derivative rules that would apply due to the speed of constant will apply the upset. So let's go ahead and take the derivative of this eso. We would write f sub X to represent the partial with respect tax. Now we have two functions being multiplied together. So we use product rule. So this will be three. Execute del by Dell X of X Square plus y squared race to the fourth and then plus X squared plus y squared raise the fourth del by Dell X of three x cute. Now over here for us to take this derivative will need to use chain rule, um, as well as powerful. So we first do power. So it be four x squared plus y square now to the cube and then we have to take the derivative of insight here. So we have X squared, which is going to be two X. But now remember, we're assuming why Squared is a constant. So this is just going to be plus zero now over here to take the derivative of three execute. Well, we would just do what we would normally do in one dimensional calculus. Just use power. Also be nine x script. Uh, and then let's see if we can, like, factor anything out to maybe make this look a little bit prettier. So, actually, I'll just go out and multiply everything together person that we could do that. Eso we have 12, and then this will be or actually not hold, um, 24 x to the fourth, and then we have X square plus y squared, cubed and then plus X square plus y squared to the fourth on, then just nine x squared. So it looks like actually, let me move this nine out front. We can factor out a three from here. It will be three times, and then we could factor out an X squared from here. Um and then this to a power of three. And then after that, we would be left with so would be eight x squared. And then the rest of that's gone. And then plus three s o that expert is gone and they would just be X word plus y squared. Then we could go ahead and distribute the three. So we get three x squared X squared plus y squared, cubed. Um, so this would give us three x squared plus three y squared, And then we can add the experts together. So the 11 expert plus three y squared. So then this here is our partial with respect to X. So last part really not needed to kind of simplify it all down. We could have just left it how we had it before But this does look a little bit prettier. So that's why I went ahead and rewrote it like this. Um and then next let me come up here, Skip this down. If you want to take the partial of this with respect to why, so do tell Beidle why this X and this extra both constants. So we can just go ahead and first pull this three execute out. So let me go ahead and write that so f sub y is equal to so be three x cube del by del y of X squared plus y squared race to the fourth. And now we can go ahead and use chain rule for this. So the three x cubed eso power rule says moved the four outfront x squared plus y squared erase the third power And then we have to take the derivative of what's on the inside. But now we're treating this X Here is a constant. So the zero plus two why they would go ahead and multiply everything together so three times, four times too. So that's 24 x cubed. And we have, um, actually let me with that why? They're so 24 x cubed times why and then X squared plus y squared. Cute. So this here is going to be our partial or why? Um, yeah, partial derivative with respect to why

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