I came across the following question and am uncertain whether my result can be further developed (I don't really see the interest in the question, if my answer is the final form - it doesn't seem particularly revealing, as a question). Am I missing anything?
Let $w = f(x,y,z)$ be a function subject to $xy = kz$, where $k$ is a constant. Find $\big(\frac{\partial w}{\partial z}\big)_x$.
My attempt:
$\big(\frac{\partial w}{\partial z}\big)_x$ implies that we have $w = w(x,z)$ and $y = y(x,z)$. Applying the chain rule, we have:
$$ \Big(\frac{\partial w}{\partial z}\Big)_x % = \Big(\frac{\partial f}{\partial y}\Big) \Big(\frac{\partial y}{\partial z}\Big)_x % + \Big(\frac{\partial f}{\partial z}\Big), $$ where, for $x \neq 0$, $y = k \tfrac{z}{x}$, so $\Big(\frac{\partial y}{\partial z}\Big)_x = \frac{k}{x}$, as such:
$$\Big(\frac{\partial w}{\partial z}\Big)_x % = \Big(\frac{\partial f}{\partial y}\Big) \frac{k}{x} % + \Big(\frac{\partial f}{\partial z}\Big).$$