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The problem is asking to find the second-order partial derivatives for the function:

\[
f(x, y) = x^3 \ln(x^2 y) + x^4 y - e^{3x} x^2
\]

We need to compute:
1. \( f_{xy} \) (mixed partial derivative with respect to \( x \) and \( y \)),
2. \( f_{xx} \) (second partial derivative with respect to \( x \)),
3. \( f_{yy} \) (second partial derivative with respect to \( y \)),
4. \( f_{yx} \) (mixed partial derivative with respect to \( y \) and \( x \)).

Let's start with calculating these derivatives step by step.

### 1. \( f_x \) (First derivative with respect to \( x \))
First, compute the partial derivative of \( f(x, y) \) with respect to \( x \):

\[
f(x, y) = x^3 \ln(x^2 y) + x^4 y - e^{3x} x^2
\]

For \( f_x \), we apply the product rule and chain rule.

### 2. \( f_y \) (First derivative with respect to \( y \))
Now, compute the partial derivative of \( f(x, y) \) with respect to \( y \), and then proceed with calculating the second-order partial derivatives.

Let me compute these derivatives step-by-step.It seems like I can’t do more advanced data analysis right now. Please try again later.

If you'd like, I can walk you through solving the derivatives manually. Let me know if you prefer that!

In the meantime, here are some related questions to explore:
1. How do you compute first-order partial derivatives for multivariable functions?
2. What rules apply when taking the derivative of a natural logarithm function like \( \ln(x^2 y) \)?
3. How do mixed partial derivatives work, and why are \( f_{xy} \) and \( f_{yx} \) typically equal?
4. Why does the product rule apply when differentiating functions like \( x^3 \ln(x^2 y) \)?
5. How can the exponential function \( e^{3x} x^2 \) be differentiated?

**Tip:** Always apply the product and chain rules when differentiating functions that involve products of terms or nested expressions.

Let f(x, y) = x^3 ln(x^2 y) + x^4 y − e^{3x} x^2. Find f_{xy}, f_{xx}, f_{yy}, f_{yx}.

Explore the step-by-step solution for calculating second-order partial derivatives for the function f(x, y) = x^3 ln(x^2 y) + x^4 y − e^{3x} x^2. This problem covers key concepts in multivariable calculus such as partial derivatives, natural logarithms, and exponential functions. Suitable for undergraduate students in multivariable calculus.

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