To solve this problem, we need to find the partial derivatives $f_x(x, y)$, $f_y(x, y)$, $f_{xy}(x, y)$, and $f_{yx}(x, y)$ of the function:

$f(x, y) = x y e^{1/y}$

Step 1: Calculate $f_x(x, y)$

The partial derivative of $f$ with respect to $x$ is found by differentiating $f(x, y) = x y e^{1/y}$ with respect to $x$, treating $y$ as a constant.

$f_x(x, y) = \frac{\partial}{\partial x} \left( x y e^{1/y} \right) = y e^{1/y}$

Step 2: Calculate $f_y(x, y)$

The partial derivative of $f$ with respect to $y$ is found by differentiating $f(x, y) = x y e^{1/y}$ with respect to $y$, treating $x$ as a constant.

To differentiate with respect to $y$, we apply the product rule, as $f(x, y) = x \cdot (y e^{1/y})$:

$f_y(x, y) = x \cdot \frac{d}{dy}\left(y e^{1/y}\right)$

Now, differentiate $y e^{1/y}$ with respect to $y$:

$\frac{d}{dy}\left(y e^{1/y}\right) = e^{1/y} + y \cdot \frac{d}{dy}\left(e^{1/y}\right)$

Since $\frac{d}{dy}\left(e^{1/y}\right) = e^{1/y} \cdot \left(-\frac{1}{y^2}\right)$:

$= e^{1/y} - \frac{e^{1/y}}{y} = e^{1/y} \left(1 - \frac{1}{y}\right)$

Thus,

$f_y(x, y) = x \cdot e^{1/y} \left(1 - \frac{1}{y}\right) = x e^{1/y} \left(1 - \frac{1}{y}\right)$

Step 3: Calculate $f_{xy}(x, y)$

The mixed partial derivative $f_{xy}(x, y)$ is the partial derivative of $f_x(x, y) = y e^{1/y}$ with respect to $y$.

$f_{xy}(x, y) = \frac{\partial}{\partial y} \left(y e^{1/y}\right)$

As computed earlier:

$f_{xy}(x, y) = e^{1/y} \left(1 - \frac{1}{y}\right)$

Step 4: Calculate $f_{yx}(x, y)$

Since mixed partial derivatives are usually equal if the function is continuous and well-behaved, we have:

$f_{yx}(x, y) = f_{xy}(x, y) = e^{1/y} \left(1 - \frac{1}{y}\right)$

Summary of Results

1. $f_x(x, y) = y e^{1/y}$
2. $f_y(x, y) = x e^{1/y} \left(1 - \frac{1}{y}\right)$
3. $f_{xy}(x, y) = e^{1/y} \left(1 - \frac{1}{y}\right)$
4. $f_{yx}(x, y) = e^{1/y} \left(1 - \frac{1}{y}\right)$

Would you like further details or have any questions?

Related Questions:

1. How do we determine if mixed partial derivatives are equal?
2. What is the significance of the exponential function $e^{1/y}$ in partial derivatives?
3. How does the product rule apply in partial differentiation?
4. What happens if $y = 0$ in this function?
5. Can you apply these derivatives to analyze the behavior of $f(x, y)$ near specific points?

Tip: Always check for continuity when assuming mixed partial derivatives are equal.