To compute the second partial derivative $\frac{\partial^2 f}{\partial y \partial x}$, or $\frac{\partial^2 f(x, y)}{\partial x \partial y}$, we follow these general steps:

1. Start with the function $f(x, y)$:

Begin by identifying the function $f(x, y)$. For example, the document provides the function: $f(x, y) = (x + y - 2)^2 + e^{x^2 - 2y^2 - xy}$

2. Compute the first partial derivative with respect to $x$:

The first partial derivative of $f$ with respect to $x$, denoted $\frac{\partial f}{\partial x}$, involves applying the differentiation rules to the function while treating $y$ as a constant: $\frac{\partial f}{\partial x} = 2(x + y - 2) + (2x - y)e^{x^2 - 2y^2 - xy}.$

3. Differentiate $\frac{\partial f}{\partial x}$ with respect to $y$:

Next, take the partial derivative of $\frac{\partial f}{\partial x}$ with respect to $y$. This involves differentiating each term in $\frac{\partial f}{\partial x}$ while treating $x$ as constant: $\frac{\partial}{\partial y} \left[ 2(x + y - 2) + (2x - y)e^{x^2 - 2y^2 - xy} \right].$ You'll need to apply the product rule to the exponential term and differentiate each term accordingly.

4. Obtain $\frac{\partial^2 f}{\partial y \partial x}$:

After performing the differentiation in Step 3, the result will be $\frac{\partial^2 f}{\partial y \partial x}$, the second mixed partial derivative.

Practical Application in the Document:

In the provided document, the partial derivatives for the function $f(x, y)$ have already been computed at certain points. For example, at the point $(2, 1)$, the mixed second partial derivative is calculated as: $\frac{\partial^2 f}{\partial y \partial x}(2, 1) = -14.$

This suggests that after performing the required differentiation, the specific value of the mixed partial derivative at $(2, 1)$ was determined as $-14$.

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Would you like more details on the differentiation steps, or do you have any further questions?

Here are 5 questions to expand on this:

1. How do we apply the chain rule when differentiating complex expressions like exponentials?
2. What is the significance of mixed partial derivatives in multivariable calculus?
3. How does the order of differentiation affect the result of mixed partial derivatives?
4. What conditions ensure that $\frac{\partial^2 f}{\partial y \partial x} = \frac{\partial^2 f}{\partial x \partial y}$?
5. How do partial derivatives contribute to finding the tangent plane to a surface?

Tip: Ensure clarity in applying product and chain rules when working with mixed partial derivatives, especially for functions involving exponentials or powers.