Let's express the partial derivatives of $w$ with respect to $u$ and $y$, denoted as $\frac{\partial w}{\partial u}$ and $\frac{\partial w}{\partial y}$, as functions of $u$ and $y$.

To do this, we need the explicit relationship between $w$, $u$, and $y$. Without this, I can outline general possibilities based on different cases, but typically:

1. If $w$ is a function of $u$ and $y$, i.e., $w = f(u, y)$, then:

   • $\frac{\partial w}{\partial u} = \frac{\partial f}{\partial u}$
   • $\frac{\partial w}{\partial y} = \frac{\partial f}{\partial y}$

2. If $u$ and $y$ are functions of other variables, say $x$, then you might use the chain rule:

   • $\frac{\partial w}{\partial u} = \frac{\partial w}{\partial x} \cdot \frac{\partial x}{\partial u}$
   • $\frac{\partial w}{\partial y} = \frac{\partial w}{\partial x} \cdot \frac{\partial x}{\partial y}$

Would you mind providing the function or equation that links $w$ with $u$ and $y$ for more precise expressions?

Questions:

1. What specific function relates $w$, $u$, and $y$?
2. Are there other variables involved in this expression?
3. Is there a chain rule application in the context of this question?
4. Are $u$ and $y$ independent or dependent variables in your scenario?
5. Is this part of a larger multivariable calculus problem?

Tip:

When working with partial derivatives, always identify whether the variables involved are dependent on other variables to properly apply the chain rule.