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Let's analyze the new problem shown in the image. The problem seems to involve partial derivatives and mixed second-order derivatives of a function \( f(x, y) \).

The problem asks us to identify the correct expression for a given derivative of a function. Based on the options and common types of questions involving partial derivatives, it looks like you need to find the mixed partial derivative \(\frac{\partial^2 f}{\partial x \partial y}\).

### Solution Process

Given a function \( f(x, y) \), the mixed partial derivative \(\frac{\partial^2 f}{\partial x \partial y}\) means we first take the partial derivative of \( f \) with respect to \( y \), and then take the derivative of the result with respect to \( x \).

Let's look at the steps typically involved:

1. **First Partial Derivative with respect to \( y \)**:
   - Compute \( \frac{\partial f}{\partial y} \) while treating \( x \) as a constant.

2. **

Learn how to solve the problem involving mixed partial derivatives of a function in multivariable calculus. This problem requires computing the mixed partial derivative \(\frac{\partial^2 f}{\partial x \partial y}\) and understanding its application. Suitable for college-level students and those studying advanced calculus topics.

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