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

Given the function \( f(x, y) = e^{xy} \sin x \), we need to compute all the second partial derivatives. Specifically, we will find \( f_{xx} \), \( f_{yy} \), and \( f_{xy} \).

Let's go through the steps:

### Step 1: Compute the First Partial Derivatives
1. **First partial derivative with respect to \( x \):**
   \[
   f_x = \frac{\partial}{\partial x}(e^{xy} \sin x)
   \]
   Using the product rule:
   \[
   f_x = e^{xy} \frac{\partial}{\partial x}(\sin x) + \sin x \frac{\partial}{\partial x}(e^{xy})
   \]
   \[
   f_x = e^{xy} \cos x + \sin x \cdot y e^{xy}
   \]
   Simplifying, we get:
   \[
   f_x = e^{xy} (\cos x + y \sin x)
   \]

2. **First partial derivative with respect to \( y \):**
   \[
   f_y = \frac{\partial}{\partial y}(e^{xy} \sin x)
   \]
   Here, we treat \( \sin x \) as a constant with respect to \( y \):
   \[
   f_y = \sin x \cdot \frac{\partial}{\partial y}(e^{xy})
   \]
   \[
   f_y = \sin x \cdot x e^{xy} = x e^{xy} \sin x
   \]

### Step 2: Compute the Second Partial Derivatives
1. **Second partial derivative with respect to \( x \):**
   \[
   f_{xx} = \frac{\partial}{\partial x}(f_x) = \frac{\partial}{\partial x} \left( e^{xy} (\cos x + y \sin x) \right)
   \]
   Applying the product rule:
   \[
   f_{xx} = e^{xy} \frac{\partial}{\partial x}(\cos x + y \sin x) + (\cos x + y \sin x) \frac{\partial}{\partial x}(e^{xy})
   \]
   \[
   f_{xx} = e^{xy} (-\sin x + y \cos x) + (\cos x + y \sin x) \cdot y e^{xy}
   \]
   Simplifying, we get:
   \[
   f_{xx} = e^{xy} \left(-\sin x + y \cos x + y \cos x + y^2 \sin x \right)
   \]
   \[
   f_{xx} = e^{xy} \left(y^2 \sin x + 2y \cos x - \sin x \right)
   \]

2. **Second partial derivative with respect to \( y \):**
   \[
   f_{yy} = \frac{\partial}{\partial y}(f_y) = \frac{\partial}{\partial y}(x e^{xy} \sin x)
   \]
   \[
   f_{yy} = x \sin x \cdot \frac{\partial}{\partial y}(e^{xy}) = x \sin x \cdot x e^{xy}
   \]
   \[
   f_{yy} = x^2 e^{xy} \sin x
   \]

3. **Mixed partial derivative with respect to \( x \) and \( y \):**
   \[
   f_{xy} = \frac{\partial}{\partial y}(f_x) = \frac{\partial}{\partial y} \left( e^{xy} (\cos x + y \sin x) \right)
   \]
   Applying the product rule:
   \[
   f_{xy} = e^{xy} \cdot \sin x + (\cos x + y \sin x) \cdot x e^{xy}
   \]
   Simplifying:
   \[
   f_{xy} = e^{xy} \sin x + x e^{xy} (\cos x + y \sin x)
   \]
   \[
   f_{xy} = e^{xy} (\sin x + x \cos x + xy \sin x)
   \]

### Summary of Second Partial Derivatives
- \( f_{xx} = e^{xy} (y^2 \sin x + 2y \cos x - \sin x) \)
- \( f_{yy} = x^2 e^{xy} \sin x \)
- \( f_{xy} = e^{xy} (\sin x + x \cos x + xy \sin x) \)

Would you like further details on each step, or have any specific questions?

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### Additional Questions
1. How would the process differ if \( f(x, y) = e^{xy} \cos x \)?
2. What if we needed only the mixed derivative \( f_{xy} \)?
3. Could you compute third partial derivatives of this function?
4. How does changing \( e^{xy} \) to \( e^{x+y} \) affect the result?
5. How would this approach change for \( f(x, y, z) = e^{xyz} \sin x \)?

### Tip:
Always verify mixed derivatives \( f_{xy} = f_{yx} \) to ensure correct partial differentiation in multi-variable functions.

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

Explore the computation of all second partial derivatives for the function f(x, y) = e^xy sin x. This solution provides a step-by-step guide to finding f_x, f_y, f_{xx}, f_{yy}, and the mixed partial derivative f_{xy}. Suitable for university-level calculus students.

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