Help with a Gradient problem for implicit equation

Question

Consider the functions $(,)=^3$, where $(,)$ is defined implicitly by the equation $^2+^2=5$. Compute the gradient of $w(x,y)$ when $=2$ and $\frac{∂}{∂}=0$.

Answer 1

$$\frac{\partial w}{\partial y} = x^3+3x^2y\frac{\partial x}{\partial y} $$

As $x = x(y,z)$, we have $$x^2 + 2xy\frac{\partial x}{\partial y} + z^2\frac{\partial x}{\partial y} = 0$$

So, as $\frac{\partial w}{\partial y} = 0$, we get

\begin{align} x^3 &= \frac{3x^4y}{z^2+2xy} \\ \implies z^2 &= xy \\ \end{align}

So, we have $x^2y+x^2y=5$, which with $x=2$ gives $y=\frac{5}{8}$, $z = \frac{\sqrt 5}{2}$.

We can calculate $\frac{\partial w}{\partial x}$ similarly.