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Find $ dy/dx $ by implicit differentiation.

$ \tan^{-1} (x^2y) = x + xy^2 $

$$y^{\prime}=\frac{1+y^{2}-\frac{2 x y}{1+x^{4} y^{2}}}{\frac{x^{2}}{1+x^{4} y^{2}}-2 x y} \text { or } y^{\prime}=\frac{1+x^{4} y^{2}+y^{2}+x^{4} y^{4}-2 x y}{x^{2}-2 x y-2 x^{5} y^{3}}$$

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for this problem, we're gonna be doing inputs of differentiation, which is incredibly useful when we have excess and wise, um, in an equation or we have two functions being set equal to each other. So how it works is we're gonna take the derivative of both sides. We have the inverse tangent right here. The inverse tangent of X squared. Why, equal to X plus X y squared? I will take the derivative of both sides. Um, when we take the derivative of the left side, we're gonna have to use the chain rule. We know that what we'll get is, uh, going to end up being since the derivative of the inverse tangent is 1/1 plus U squared, for example, we'll end up getting is, in this case, 1/1 plus x squared. Why? But then we have to do the chain rule. So it's gonna be times on then everything in here. So it's gonna be X squared. This will be the product drawer here. So X squared kind of my prime. Plus why times two x, a chain right there. And then we take the dirt of this so we'll get one because the derivative of X. Just one eso We'll have one plus and then off to do change rules. That will be X times two. Why? Plus why squared plus two i times why prime plus y squared thought that we end up getting here. As a result, we can multiply everything through right here. So what we end up having as result, is going to be X squared over this portion down here. And then, um, that will be in this case since we had to square this portion. Since we have to do you squared? This was squared right here. So what? This is actually gonna be X to the fourth. Why squared? And then we'll have minus two X. Why, Onda? Another thing we have to keep in mind is that in having the minus two x y, this is going to be all times de y dx, which we can write his wide crime. I will be equal to one plus y squared minus and then two x y over one plus x to the force wide squared. So the reason we get this is because we know that, um, we can combine these on dwhite. We have the negative two x y This is coming from our distribution right here, the white Times to access to X y. And we know that it will end up becoming, um, minus that value when we subtract it because we have two. X y here is well than what we can dio. To simplify things further is we're going to divide both sides by this portion right here. So what will end up getting as a result when we divide all this is we're going to have that why prime is equal to one plus X to the fourth wide squared plus why squared US X to the fourth. Why to the fourth minus two x y, and that is all going to be divided by X squared minus two x y minus two x to the fifth y Cube. And that will be our final answer for why Prime

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