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

(a) $2 y e^{3 x y}(3 x y+1)$(b) $2 x e^{3 x y}(3 x y+1)$

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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01:49

Find the first partial der…

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00:55

So in order for us to find the partials with respect to both x and y of this, nor will we end up doing is we treat all of the other variables as if they're constants, and then we just take the derivative like we would in the one variable case. So let's come over here and first do the partial with respect. X o Dell by Dell X It's on the left. We would normally write. This adds, um f sub x. So the partial with respect to X um And then we're going to treat this Why? And this Why? Just as if they were Constance. So at first we could just go ahead and pull out, too. Why, from this would be to i del by Dell x of x times e to the three x. Why, Yeah, And now we have a function that depends on X multiplied by a function that also depends on X. So we use product rule. So would be to why, uh, times so it be Extell by Dell X of E to the three x y, and then plus E to the three x y del by Dell X Now to take the derivative of E 23 x y. So we first take the derivative of the outside because we need to use change rules. So that doesn't change to those still be e 23 x y But then, remember, change will just take the derivative of the inside and we're going to treat this. Why is a constant So it's like three times Why both? Constant. So we just take the derivative of X essentially, which will give us three. Why? Andan over here taking the derivative of X just gives us one. So let's go ahead and raped this out. So you have to y times it looks like three x y AII to the three x y plus e to the three x y And then we could go ahead and factor out that three to the ex wife because there is a common factor. I mean, not really needed, but just to clean it up a little bit. So we have to i iii to the three x y And then times, um three x Y plus one. Yeah. Uh huh. And so this is our partial with respect to X right? Next, we can go ahead and take partial with respect to why eso them come appear and pull this down. And so now, instead of treating these wise is Constance, we treat the wise is our variables that we're going to treat the exes, as are constants. So we do, Del by Dell X off this. So you're not dealt by dogs? Where did that Dell? By don't Why of that? So we have f sub y or the partial of f with respect to buy is able to remember this x and this expatriates constants. So let's just pull it out just like we did before the two x tell by del y of y AII to the three x times wife. And again we have a function that depends on why multiply function. That depends on why. So we use product raw. So this will be two x times. So why Dell? Beidle Why, uh v 23 x y and then plus E to the three x y times. Uh del Beidle, Why of why? And again this derivative here is going to be essentially the same thing over and just have an X there instead. Um, so we have e 23 x y and then we take the road of the inside Do the chain rule eso the X now is a constant so we just take the derivative wise would be three x And then over here, del by del Y y is just one And now we go ahead, multiply everything together as we have two X um Times three x y e to the three x y plus e +23 ex wife that we could go ahead and factor out the ease that the X y so we have to x e to the three x y and then times three x y plus one. So then this is going to be our partial with respect to why, okay?

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