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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.

Learn how to compute the expression x * Partial f / Partial x + y * Partial f / Partial y using logarithm properties and partial derivatives. This problem covers multivariable calculus concepts and is suitable for undergraduate students.

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