The document contains a set of practice problems for partial derivatives, chain rules, homogeneous functions, and applications of Euler's theorem. Here’s a summary of a few selected questions along with their main objectives:

1. Compute Partial Derivatives - Problems like finding $\frac{\partial u}{\partial x}$, $\frac{\partial u}{\partial y}$, and $\frac{\partial u}{\partial z}$ for functions such as $u = \log(x^3 + y^3 + z^3 - 3xyz)$.

2. Verification of Partial Derivatives Properties - Demonstrating identities like $\frac{\partial u}{\partial x} + \frac{\partial u}{\partial y} + \frac{\partial u}{\partial z} = \frac{3}{x + y + z}$ under specific functions.

3. Implicit Differentiation Applications - Solving for partial derivatives when functions are given implicitly, such as when $\tan u = \frac{x}{y}$ or $\sin u = \frac{x}{y}$.

4. Higher-Order Derivatives - Calculating second-order partial derivatives, for example, $\frac{\partial^2 u}{\partial x^2}$ for functions like $u = \log(y \sin x + x \sin y)$.

5. Euler's Theorem and Homogeneous Functions - Verifying and applying Euler's theorem, such as proving that for a function $u = f(y/x)$, the identity $x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} = 0$ holds.

If you would like detailed solutions for specific questions or concepts explained further, feel free to ask!

Here are five questions to expand on the topics:

1. How do you find partial derivatives for implicit functions?
2. What are homogeneous functions, and how do they relate to Euler's theorem?
3. How is implicit differentiation applied in partial derivatives?
4. Why are second-order partial derivatives important in multivariable calculus?
5. Can you explain the application of Euler’s theorem in economics or physics?

Tip: When working with implicit functions, remember to differentiate each term with respect to the variable of interest, treating other variables as constants.