To compute the first-order partial derivatives of the function $u = \arctan\left(\frac{x^2 + y^2}{x + y}\right)$ with respect to $x$ and $y$, we follow the steps below:

Function:

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

We'll compute the partial derivatives $\frac{\partial u}{\partial x}$ and $\frac{\partial u}{\partial y}$.

Step 1: Differentiate $u(x, y)$ with respect to $x$

We use the chain rule, noting that the derivative of $\arctan(z)$ is $\frac{1}{1 + z^2}$. Let $z = \frac{x^2 + y^2}{x + y}.$ Then, $\frac{\partial u}{\partial x} = \frac{1}{1 + z^2} \cdot \frac{\partial z}{\partial x}.$

Now, differentiate $z$ with respect to $x$. Using the quotient rule, the derivative of $z = \frac{x^2 + y^2}{x + y}$ is: $\frac{\partial z}{\partial x} = \frac{(2x)(x + y) - (x^2 + y^2)(1)}{(x + y)^2}.$ Simplifying the numerator: $\frac{\partial z}{\partial x} = \frac{2x(x + y) - (x^2 + y^2)}{(x + y)^2} = \frac{2x^2 + 2xy - x^2 - y^2}{(x + y)^2} = \frac{x^2 + 2xy - y^2}{(x + y)^2}.$

Thus, $\frac{\partial u}{\partial x} = \frac{1}{1 + z^2} \cdot \frac{x^2 + 2xy - y^2}{(x + y)^2}.$

Step 2: Differentiate $u(x, y)$ with respect to $y$

Again, we apply the chain rule: $\frac{\partial u}{\partial y} = \frac{1}{1 + z^2} \cdot \frac{\partial z}{\partial y}.$

Now differentiate $z = \frac{x^2 + y^2}{x + y}$ with respect to $y$. Again, use the quotient rule: $\frac{\partial z}{\partial y} = \frac{(2y)(x + y) - (x^2 + y^2)(1)}{(x + y)^2}.$ Simplifying the numerator: $\frac{\partial z}{\partial y} = \frac{2y(x + y) - (x^2 + y^2)}{(x + y)^2} = \frac{2xy + 2y^2 - x^2 - y^2}{(x + y)^2} = \frac{-x^2 + 2xy + y^2}{(x + y)^2}.$

Thus, $\frac{\partial u}{\partial y} = \frac{1}{1 + z^2} \cdot \frac{-x^2 + 2xy + y^2}{(x + y)^2}.$

Final Partial Derivatives

$\frac{\partial u}{\partial x} = \frac{1}{1 + \left(\frac{x^2 + y^2}{x + y}\right)^2} \cdot \frac{x^2 + 2xy - y^2}{(x + y)^2}$ and $\frac{\partial u}{\partial y} = \frac{1}{1 + \left(\frac{x^2 + y^2}{x + y}\right)^2} \cdot \frac{-x^2 + 2xy + y^2}{(x + y)^2}.$

Would you like more details on any of the steps or further explanation?

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Related Questions:

1. How do you apply the quotient rule in partial derivatives?
2. What is the chain rule for functions of multiple variables?
3. How do you compute second-order partial derivatives for this function?
4. How would you simplify the expression for $z$ in this context?
5. What are the applications of the arctangent function in multivariable calculus?

Tip:

When differentiating functions involving fractions, the quotient rule is essential, but sometimes expressions can be simplified before differentiation to reduce complexity.