 Find all second partial derivatives of f(x, y) = e
xy sin x. fxy

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.

Find all second partial derivatives of f(x, y) = e^(xy) sin x. fxy

Explore the process of finding second partial derivatives of the function f(x, y) = e^(xy) sin x, including mixed partial derivatives and the application of the product rule. This detailed solution covers concepts in calculus and is suitable for college-level students.

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