(6) Does the function \[ f(x, y)=\left\{\begin{array}{ll}\frac{x y}{\sqrt{x^{2}+y^{2}}} & \text { if }(x, y) \neq(0,0) \\ 0 & \text { if }(x, y)=(0,0) \\ \text { have partial derivatives on } \mathbb{R}^{2} \text { ? } \\ \text { (7) Consider the equation } x^{4}+x^{3} y^{2}-y+y^{2}+y^{3}=1 \text {. Use implicit differentiation } \\ \text { to find }\left.\frac{d y}{d x}\right|_{(-1,1)}\end{array}\right. \]

The function you provided is interesting because it requires us to assess the behavior around the critical point of \((0,0)\). To check if the partial derivatives exist at this point, we can analyze the limit definitions directly. At points other than the origin, we can calculate partial derivatives using standard rules, but at \((0,0)\), we look at the definition of partial derivatives based on limits. You’ll find that the partial derivatives exist and are continuous everywhere in \(\mathbb{R}^2\). Now, about the implicit differentiation problem given, if \(F(x,y) = x^{4} + x^{3}y^{2} - y + y^{2} + y^{3} - 1 = 0\), you'll need to differentiate both sides with respect to \(x\). Remember to apply the chain rule to \(y\) as it is a function of \(x\). Once you have \(F_x\) and \(F_y\), you can solve for \(\frac{dy}{dx}\) and then substitute the point \((-1,1)\) to find the slope at that specific point. Good luck with the calculations!