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The problem provided in the image is:

Given \( f(x, y) = x^2 + x^2y - 2y^2 \), evaluate \( f_x \) and \( f_y \).

### Solution:

To solve this, we will compute the **partial derivatives** of \( f(x, y) \) with respect to \( x \) and \( y \).

#### 1. \( f_x(x, y) \) — Partial derivative of \( f \) with respect to \( x \):

We differentiate \( f(x, y) = x^2 + x^2y - 2y^2 \) with respect to \( x \) while treating \( y \) as a constant:

\[
f_x(x, y) = \frac{\partial}{\partial x} \left( x^2 + x^2y - 2y^2 \right)
\]

- The derivative of \( x^2 \) with respect to \( x \) is \( 2x \).
- The derivative of \( x^2y \) with respect to \( x \) is \( 2xy \) (since \( y \) is treated as a constant).
- The derivative of \( -2y^2 \) with respect to \( x \) is \( 0 \) (as there is no \( x \) involved in this term).

So, 
\[
f_x(x, y) = 2x + 2xy
\]

#### 2. \( f_y(x, y) \) — Partial derivative of \( f \) with respect to \( y \):

Now, we differentiate \( f(x, y) = x^2 + x^2y - 2y^2 \) with respect to \( y \), treating \( x \) as a constant:

\[
f_y(x, y) = \frac{\partial}{\partial y} \left( x^2 + x^2y - 2y^2 \right)
\]

- The derivative of \( x^2 \) with respect to \( y \) is \( 0 \) (as there is no \( y \) involved).
- The derivative of \( x^2y \) with respect to \( y \) is \( x^2 \).
- The derivative of \( -2y^2 \) with respect to \( y \) is \( -4y \).

So,
\[
f_y(x, y) = x^2 - 4y
\]

### Final Results:
- \( f_x(x, y) = 2x + 2xy \)
- \( f_y(x, y) = x^2 - 4y \)

Would you like further details or explanations on any step?

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Here are five related questions to explore:

1. How do you compute second-order partial derivatives for this function?
2. What happens if you evaluate \( f_x \) and \( f_y \) at specific points like \( (1, 1) \)?
3. Can you use these partial derivatives to find the gradient of the function?
4. How would you interpret these partial derivatives in terms of the function's behavior?
5. What are the higher-order derivatives of this function?

**Tip:** Remember that partial derivatives give you the rate of change of the function in the direction of each variable while holding the other constant.

Given f(x, y) = x^2 + x^2y - 2y^2, evaluate f_x and f_y.

Learn how to compute the partial derivatives of f(x, y) = x^2 + x^2y - 2y^2 with respect to x and y. This solution covers key concepts in multivariable calculus and differentiation, suitable for students in advanced high school or early college levels.

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