Functions of two variables and partial derivatives practice question 12

Here $z$ is not given by a formula; the equation ties it to $x$ and $y$. Which technique handles that, and which variable stays fixed while you differentiate in $x$?

Differentiate both sides in $x$, holding $y$ fixed and treating $z$ as a function of $x$ and $y$: every term containing $z$ produces a factor $\dfrac{\partial z}{\partial x}$.

You should get $2xz + x^{2}\dfrac{\partial z}{\partial x} + 3yz^{2}\dfrac{\partial z}{\partial x} + y = 0$. Substitute $(2, 1, 1)$, then solve.