You are correct that . However, we will find a different answer for at the same point.
To differentiate both sides by , we simply treat as if it were a constant and as if it were a function of .
Consider the equation
as defining implicitly as a function of and . Compute the following.
zē + 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
Should (∂z/∂y) answer be the same at the x? I don't get it. Please help.