The given function is:

$f(x, y) = \ln\left(x + \sqrt{x^2 + y^2}\right)$

1. Partial derivative with respect to $x$:

We need to compute the partial derivative $\frac{\partial f}{\partial x}$.

Let’s differentiate step by step:

$f(x, y) = \ln\left(x + \sqrt{x^2 + y^2}\right)$

Using the chain rule for derivatives: $\frac{\partial}{\partial x} \ln(u) = \frac{1}{u} \cdot \frac{\partial u}{\partial x}$ where $u = x + \sqrt{x^2 + y^2}$.

Now, we compute $\frac{\partial u}{\partial x}$:

$\frac{\partial}{\partial x} \left(x + \sqrt{x^2 + y^2}\right) = 1 + \frac{1}{2} \cdot (x^2 + y^2)^{-\frac{1}{2}} \cdot 2x$

Simplifying:

$\frac{\partial}{\partial x} \left(x + \sqrt{x^2 + y^2}\right) = 1 + \frac{x}{\sqrt{x^2 + y^2}}$

Thus, the partial derivative of $f$ with respect to $x$ is:

$\frac{\partial f}{\partial x} = \frac{1}{x + \sqrt{x^2 + y^2}} \cdot \left(1 + \frac{x}{\sqrt{x^2 + y^2}}\right)$

2. Partial derivative with respect to $y$:

Now, we compute the partial derivative $\frac{\partial f}{\partial y}$.

Again, we use the chain rule. First, we differentiate $u = x + \sqrt{x^2 + y^2}$ with respect to $y$:

$\frac{\partial}{\partial y} \left(x + \sqrt{x^2 + y^2}\right) = \frac{1}{2} \cdot (x^2 + y^2)^{-\frac{1}{2}} \cdot 2y$

Simplifying:

$\frac{\partial}{\partial y} \left(x + \sqrt{x^2 + y^2}\right) = \frac{y}{\sqrt{x^2 + y^2}}$

Thus, the partial derivative of $f$ with respect to $y$ is:

$\frac{\partial f}{\partial y} = \frac{1}{x + \sqrt{x^2 + y^2}} \cdot \frac{y}{\sqrt{x^2 + y^2}}$

3. Evaluate $\frac{\partial f}{\partial x}$ at $(2, 3)$:

Substitute $x = 2$ and $y = 3$ into the expression for $\frac{\partial f}{\partial x}$:

$\frac{\partial f}{\partial x} (2, 3) = \frac{1}{2 + \sqrt{2^2 + 3^2}} \cdot \left(1 + \frac{2}{\sqrt{2^2 + 3^2}}\right)$

First, compute $\sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13}$.

Now, substitute:

$\frac{\partial f}{\partial x} (2, 3) = \frac{1}{2 + \sqrt{13}} \cdot \left(1 + \frac{2}{\sqrt{13}}\right)$

This is the final value for $\frac{\partial f}{\partial x}$ at $(2, 3)$.

Would you like a further breakdown or any clarifications?

Related questions:

1. What is the geometric interpretation of partial derivatives?
2. How do we apply the chain rule in multivariable functions?
3. Can you explain how to differentiate logarithmic functions involving multiple variables?
4. What are some common real-world applications of partial derivatives?
5. How would you find the second-order partial derivatives of the function?

Tip:

Always simplify the expressions inside the square root or logarithm before differentiating for easier computations.