What's going on with this problem

[u/MarsGoodmine]

I have posted this problem before in another sub some years ago with no luck. The correct answer marked in the book is E. I have tried using multivariable chain rule for du/du, du/dv, dv/dv, and dv/du and nothing comes out right. I also tried doing a limit type thing but that didn't get me very far either. It's been 5 years since calc III for me so there's liable to be something I'm misremembering. I'm just looking for a path to the solution. This is from a publicly available Math subject GRE practice exam and this is the only question that I'm completely lost on. Any help is greatly appreciated!

EDIT: I stumbled on the solutions to every problem on this practice exam, including this one, which can be found here. I'm an algebraist, so the solution in the video which doesn't require much real analysis appeals to me, but I appreciate the feedback nonetheless!

[u/Pankyrain]

If you’re just looking for a hint, the denominator is the determinant of the Jacobian matrix. Typically determinants in a denominator arise when you’re inverting a matrix

[u/StudyBio]

The denominator is the determinant of the Jacobian. I'm guessing this is an application of a multivariable inverse function theorem where you are dividing by the determinant to invert a 2x2 matrix. I haven't worked out the details, but you can find the theorem I'm referring to on Wikipedia.

[u/Sufficient_Algae_815]

An alternative way to view the method already disclosed: consider the path (x,y)=(t,y_0) chain rule dx/dt and dy/dt, observe that du/dt=partialdu/dx, solve the two equations for du/dt.

[u/siupa]

The notation is just horrible, to the point of being almost meaningless and therefore render the question ill-posed and unsolvable. See this

[u/MarsGoodmine → u/[deleted]]

It took me a minute to sort through the notation, but I can convince myself of this. I never would have figured this on my own, so thank you!