The given expression is:

\[
z = 5x \ln(x^2 + y)
\]

### To find derivatives or perform other operations, please clarify which you would like to do:

1. **Partial Derivatives**: Are you looking for \( \frac{\partial z}{\partial x} \) or \( \frac{\partial z}{\partial y} \)?
2. **Simplifications**: Do you want to simplify or rewrite the expression?
3. **Evaluations**: Do you want to evaluate it for specific values of \( x \) and \( y \)?
4. **Other Operations**: Are you looking for the gradient, Hessian matrix, or something else?

Let me know your preference, and I'd be happy to proceed!

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Here are some related questions that build on this topic:

1. How do you find the partial derivative of \( z \) with respect to \( x \)?
2. How does the chain rule apply when differentiating logarithmic functions?
3. What are the necessary steps to evaluate \( z \) when specific values for \( x \) and \( y \) are provided?
4. What is the gradient of \( z \) and how is it interpreted geometrically?
5. How do you differentiate a product of functions, such as \( 5x \ln(x^2 + y) \)?

**Tip**: When differentiating logarithmic expressions, always apply the chain rule to handle the argument of the logarithm.

Explore how to find the partial derivatives of the function z = 5x ln(x^2 + y). This problem delves into calculus concepts such as partial derivatives and the chain rule, making it suitable for advanced high school or college students.

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