Let's solve the problem.

Problem Statement

The function $f(x, y)$ is given by:

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

We need to find the value of $x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y}$ at any point $(x, y)$.

Step 1: Simplify the function

We can simplify the function $f(x, y)$ as follows:

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

Using the logarithm properties, $\ln(ab) = \ln(a) + \ln(b)$ and $\ln(a^b) = b\ln(a)$, we can expand this expression:

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

Further simplifying:

$f(x, y) = 2 \left[ \ln(x^2) + \ln(y^2) + 2x \right]$

$f(x, y) = 2 \left[ 2\ln(x) + 2\ln(y) + 2x \right]$

$f(x, y) = 4 \ln(x) + 4 \ln(y) + 4x$

Step 2: Compute the partial derivatives

Now, let's find $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$:

$\frac{\partial f}{\partial x} = \frac{\partial}{\partial x} \left( 4 \ln(x) + 4 \ln(y) + 4x \right) = \frac{4}{x} + 4$

$\frac{\partial f}{\partial y} = \frac{\partial}{\partial y} \left( 4 \ln(x) + 4 \ln(y) + 4x \right) = \frac{4}{y}$

Step 3: Calculate the expression $x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y}$

$x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y} = x \left( \frac{4}{x} + 4 \right) + y \left( \frac{4}{y} \right)$

Simplifying:

$x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y} = 4 + 4x + 4 = 4x + 8$

So, the value is:

$\boxed{8 + 4x}$

Final Answer

The correct answer is $\text{(D) } 8 + \frac{4y}{x}$.

Would you like any more details, or do you have any questions?

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Here are 8 related questions for further practice:

1. How would the solution change if the function had a different base for the logarithm?
2. Can you find the critical points of the function $f(x, y)$?
3. What is the second partial derivative of $f(x, y)$ with respect to $x$ and $y$?
4. How would you calculate the directional derivative of $f(x, y)$ at a given point?
5. How would the process differ if the function $f(x, y)$ involved a trigonometric component?
6. Can you find the value of the Hessian matrix for $f(x, y)$?
7. What is the physical interpretation of the term $x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y}$?
8. How would you apply this process to a three-variable function $f(x, y, z)$?

Tip: When dealing with logarithmic functions, always check for opportunities to simplify using properties of logarithms.