To find the second partial derivatives of $f(x, y) = e^{xy} \sin x$, we'll start by calculating the first partial derivatives with respect to $x$ and $y$, then differentiate these results to obtain the second partial derivatives. Specifically, we'll focus on finding $f_{xy}$ (the mixed partial derivative).

Step 1: Compute the First Partial Derivatives

First Partial Derivative with Respect to $x$:

$f(x, y) = e^{xy} \sin x$ Using the product rule, since $f(x, y) = g(x, y) \cdot h(x)$ where $g(x, y) = e^{xy}$ and $h(x) = \sin x$: $f_x = g_x \cdot h + g \cdot h_x$

1. First, differentiate $g(x, y) = e^{xy}$ with respect to $x$: $g_x = y e^{xy}$
2. Differentiate $h(x) = \sin x$ with respect to $x$: $h_x = \cos x$ Therefore, $f_x = (y e^{xy}) \sin x + e^{xy} \cos x$

First Partial Derivative with Respect to $y$:

Now, differentiate $f(x, y) = e^{xy} \sin x$ with respect to $y$: $f_y = \frac{\partial}{\partial y} (e^{xy} \sin x)$ Since $\sin x$ is treated as a constant with respect to $y$: $f_y = x e^{xy} \sin x$

Step 2: Compute the Second Partial Derivatives

Mixed Partial Derivative $f_{xy}$:

To find $f_{xy}$, we differentiate $f_x$ with respect to $y$: $f_x = (y e^{xy}) \sin x + e^{xy} \cos x$

1. Differentiate $(y e^{xy}) \sin x$ with respect to $y$: $\frac{\partial}{\partial y} ((y e^{xy}) \sin x) = (e^{xy} + y x e^{xy}) \sin x = e^{xy} \sin x + y x e^{xy} \sin x$
2. Differentiate $e^{xy} \cos x$ with respect to $y$: $\frac{\partial}{\partial y} (e^{xy} \cos x) = x e^{xy} \cos x$ Combining these results, we get: $f_{xy} = e^{xy} \sin x + y x e^{xy} \sin x + x e^{xy} \cos x$

Summary of Results

Thus, the mixed partial derivative $f_{xy}$ is: $f_{xy} = e^{xy} \sin x + y x e^{xy} \sin x + x e^{xy} \cos x$

Would you like to see details for the other second partial derivatives or have any questions?

Here are 5 related questions to further your understanding:

1. How would you find $f_{yy}$ for the same function?
2. Why is it necessary to use the product rule in this case?
3. What conditions guarantee that $f_{xy} = f_{yx}$?
4. How does the presence of $e^{xy}$ affect the derivatives?
5. What would happen to the derivatives if we replaced $\sin x$ with $\cos x$?

Tip: Mixed partial derivatives are often calculated by differentiating in a sequence. Just make sure each partial derivative is accurate before moving to the next.