The first partial derivatives of the function $w = \frac{e^v}{u + v^5}$ are:

$\frac{\partial w}{\partial u} = -\frac{e^v}{(u + v^5)^2}$

$\frac{\partial w}{\partial v} = \frac{e^v}{u + v^5} - \frac{5v^4 e^v}{(u + v^5)^2}$

Would you like further details or clarification on any step?

Here are 5 related questions to deepen your understanding:

1. How would the partial derivatives change if the denominator were a different function of $u$ and $v$?
2. What is the significance of the negative signs in these partial derivatives?
3. How would you find the second-order partial derivatives of this function?
4. What do these partial derivatives tell us about the rate of change of $w$ with respect to $u$ and $v$?
5. Could you apply the chain rule to find the derivative of $w$ with respect to some other variable that depends on $u$ and $v$?

Tip: Always remember to apply the chain rule carefully when differentiating composite functions involving multiple variables.