1. ## Partial Derivative?

Consider the equation

as defining implicitly as a function of and . Compute the following.

?

z² + 2xz(∂z/∂x) + 4y(∂z/∂x) + (2/z)(∂z/∂x) = 0
1 - 2(∂z/∂x) + 4(∂z/∂x) + 2(∂z/∂x) = 0
∂z/∂x = -1/4
2. You are correct that $\frac{\partial z}{\partial x}=-\frac{1}{4}$. However, we will find a different answer for $\frac{\partial z}{\partial y}$ at the same point.
To differentiate both sides by $y$, we simply treat $x$ as if it were a constant and $z$ as if it were a function of $y$.