If $$x=u^2-v^2$$ and $$y=2uv$$ What is $$\left(\frac{\partial u}{\partial x}\right)_y, \left(\frac{\partial u}{\partial y}\right)_x, \left(\frac{\partial v}{\partial x}\right)_y and \left(\frac{\partial v}{\partial y}\right)_x?$$
I am very unsure to think about these sorts of questions. When they are in a form where i can write f(x,y,z)=0, then i usually do it quite easily using the identity $(\frac{\partial x}{\partial z})_y =-(\frac{\partial f}{\partial z})_{x,y}/(\frac{\partial f}{\partial x})_{y,z}$, but i cant find a way to do that easily in this case. Another identity we have learned is (For f(x,y)) $df=(\frac{\partial f}{\partial x})_ydx+(\frac{\partial f}{\partial y})_xdy$, maybe this is helpful to solve it?