The function given is:

\[
u = \arctan\left(\frac{x^2 + y^2}{x + y}\right)
\]

If you need to explore or differentiate this function, let's outline a few possible steps depending on what you're looking for:

### 1. **Partial derivatives**:
   - To find the **partial derivative of \( u \)** with respect to **\( x \)** or **\( y \)**, we use the chain rule. Let’s compute these if that’s what you need.
   
### 2. **Gradient of \( u \)**:
   - The **gradient** of \( u \) will be a vector formed by \( \frac{\partial u}{\partial x} \) and \( \frac{\partial u}{\partial y} \).

### 3. **Level Curves**:
   - Level curves \( u = \text{constant} \) represent solutions to the equation \( \frac{x^2 + y^2}{x + y} = \text{constant} \), which could describe certain curves in the plane.

Would you like to proceed with any of these steps, or is there something specific you'd like to explore about this function?

### 5 Relative Questions
1. Would you like to compute the partial derivatives of \( u \) with respect to \( x \) and \( y \)?
2. Are you interested in the gradient or directional derivatives of \( u \)?
3. Would you like to analyze the behavior of \( u \) as \( x \) and \( y \) approach certain values?
4. Are you interested in visualizing the level curves of this function?
5. Would you like to apply this function to any specific values of \( x \) and \( y \)?

**Tip**: When working with functions involving arctangents, recall that the range of \( \arctan \) is \( (-\frac{\pi}{2}, \frac{\pi}{2}) \), which can help analyze the behavior of the function.

Explore how to differentiate the function u = arctan((x^2 + y^2)/(x + y)) using partial derivatives and the chain rule. This problem delves into multivariable calculus, covering key concepts like partial derivatives and gradient vectors. Suitable for college-level students in Calculus II or III.

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