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Determine the derivative.$$x^{2} e^{2 x^{3} y^{2}}-2 e^{x}+3 e^{2 y}=3 x^{3}-4 y^{2}+2 x-3 y$$

$\frac{9 x^{2}+2-6 x^{4} y^{2} e^{2 x^{3} y^{2}}-2 x e^{2 x^{3} y^{2}}+2 e^{x}-3 e^{2 y}}{4 x^{5} y e^{2 x^{3} y^{2}}+6 e^{2 y}+8 y+3}$

Algebra

Chapter 4

Exponential and Logarithmic Functions

Section 4

The Derivative of the Exponential Function

Missouri State University

Campbell University

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All right. So we're gonna do the implicit differentiation to get this answer either of the two executing y squared. This is gonna take me a minute to copy down the problem minus two e to the X. And if I have a typo, then I'll definitely get the wrong answer. Plus three e to the two y. So let's pray I copied this down correctly equals three X cubed minus four y squared. Yeah, plus two x ministry. Why? Okay, if, um even if I copy this down incorrectly, you'll get the right premise and then, you know, you just might have to fix a few of my mistakes. But anyway, um, if you notice the very first pieces a product eso we'd better do the product rule where you take the derivative of the left side which would be X squared. You leave e to the two x cubed y squared alone and then plus, now you leave X squared alone times the drift of of the right side, which should be, um, e to the two x cubed y squared. But then you have to multiply by the derivative of the exponents, which is another product rule And so you get six x squared. Leave y squared alone. Plus excuse me four x cubed. Why? But because it's implicit, you'd have to say deep, you know, times D y d x And then the next piece the derivative of two e to the X s two e to the X and then the next one because it's a chain rule, Uh, it is e to the to y, but then you have to multiply by the drift of of two y, which is too so three times to a six, do you y dx And then that equals the derivative on the right side, which should give me nine X square minus eight Y d Y d x Yeah, uh, plus two minus three d y DX. So you can already imagine why this problem is difficult as is, um, and a few other things you might notice. Um, that's, you know, multiplying this piece in distributing that piece in would be helpful to do this. Eso I'm just kind of I'm gonna do two steps at once where I need to get all my d y DX is on the same side. And, um, I will factor out because it's already long enough video eso as I'm looking at, if I distribute this, um, I don't know where this is going to end this X squared e to the two x cubed. Why squared into both those pieces? Um, you know, I also have that times the four x cubed. Why? Because that has a d y d x. Um, I also have this is it d Y d x six e to the two. Y, um I also need to add eight y to the left side, and I need to add three to the left side. Now notice I also factored out The D y d x and then on the right side is right, equals I still have nine x squared. I still have the to. So I've taken care of this that I'm underlining because I need to take care of things. I need to move a negative to the other side. So plus to eat of the X. Um, I know I wrote that I distributed this, but I've Onley distribute that to one term so far, I still have to distribute to this piece. Um, hello. Antony is subtracted over so change the sign of it. It's, um, minus may I can combine like terms as well, like six X to the fourth. Why squared e to the two x cubed y squared? But I also need to move this piece over so that becomes a minus two x e to the two x cubed. Why squared? Um, and now I'm ready to solve, which is why I went ahead and factored at the same time to show you that the soul for D y dx I need to divide that piece over So everything I haven't read stays the same sign. Well, it goes into the numerator. I guess I should say. And I'm not gonna mess with factory anything. I don't think anything even factors to X cubed. Why square? That's a terrible too, You know, the two executed y squared. And like I said, if I had a type of the beginning, I'm definitely haven't given the wrong answer. Um and then did the denominator again. I didn't combine like terms before, but I guess I could have, you know, rewriting As for X to the fourth. Why, oops, I should be next to the fifth. There's two here and three here. Why e to the two x cubed y squared, plus 6 ft of the two y plus eight y plus three. And again, nothing factors. I don't believe ending factors. This should be your answer, okay?

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