The expression in the image is the partial derivative of a function $f$ with respect to $x$, given as:

$\frac{\partial f}{\partial x} = \frac{-\sin(xy^2) \cdot y^2 \cdot (xy^2) - \cos(xy^2) \cdot y^2}{(xy^2)^2}$

Let's analyze it step by step:

Numerator:

1. The term involves two parts: $-\sin(xy^2) \cdot y^2 \cdot (xy^2) \quad \text{and} \quad -\cos(xy^2) \cdot y^2$
   • The first part is the product of the sine function, $\sin(xy^2)$, multiplied by $y^2$ and $xy^2$.
   • The second part is $-\cos(xy^2)$ multiplied by $y^2$.

Denominator:

2. The denominator is $(xy^2)^2$, which means we square $xy^2$.

Simplification Approach:

This expression could potentially be simplified further by factoring common terms in the numerator, but it would depend on the function structure.

Would you like a more detailed step-by-step simplification or further clarification on any specific part of this derivative?

Here are five possible questions based on this context:

1. What is the original function $f$ from which this derivative is obtained?
2. How can we simplify the numerator by factoring common terms?
3. What is the role of the chain rule in finding this partial derivative?
4. Can the denominator $(xy^2)^2$ be simplified in any context?
5. How would this expression behave for specific values of $x$ and $y$?

Tip: When simplifying trigonometric derivatives involving products, always look for common terms in both the numerator and denominator.