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

To find the second partial derivatives of the function \( f(x, y) = e^{xy} \sin x \), we proceed by computing the first and then the second partial derivatives with respect to \( x \) and \( y \).

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

2. **Partial derivative with respect to \( y \):**
   \[
   f_y = \frac{\partial}{\partial y}(e^{xy} \sin x)
   \]
   Differentiating with respect to \( y \), treating \( x \) as a constant:
   \[
   f_y = x e^{xy} \sin x
   \]

### Step 2: Compute the Second Partial Derivatives

1. **Second partial derivative with respect to \( x \) (i.e., \( f_{xx} \)):**
   \[
   f_{xx} = \frac{\partial}{\partial x}(f_x) = \frac{\partial}{\partial x}\left(e^{xy} \cos x + y e^{xy} \sin x\right)
   \]
   Applying the product rule to each term:
   \[
   f_{xx} = e^{xy}(-\sin x) + y e^{xy} \cos x + y \left(e^{xy} \cos x + y e^{xy} \sin x\right)
   \]
   Simplifying:
   \[
   f_{xx} = e^{xy}(-\sin x + y \cos x) + y e^{xy} \cos x + y^2 e^{xy} \sin x
   \]

2. **Mixed partial derivative with respect to \( x \) and \( y \) (i.e., \( f_{xy} \)):**
   \[
   f_{xy} = \frac{\partial}{\partial y}(f_x) = \frac{\partial}{\partial y}\left(e^{xy} \cos x + y e^{xy} \sin x\right)
   \]
   Differentiating each term with respect to \( y \):
   \[
   f_{xy} = x e^{xy} \cos x + e^{xy} \sin x + y x e^{xy} \sin x
   \]
   Simplifying:
   \[
   f_{xy} = e^{xy}(x \cos x + \sin x + xy \sin x)
   \]

3. **Second partial derivative with respect to \( y \) (i.e., \( f_{yy} \)):**
   \[
   f_{yy} = \frac{\partial}{\partial y}(f_y) = \frac{\partial}{\partial y}(x e^{xy} \sin x)
   \]
   Using the product rule:
   \[
   f_{yy} = x^2 e^{xy} \sin x
   \]

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

Let me know if you would like any additional details or have questions!

### Related Questions
1. What are higher-order partial derivatives for a function of three variables?
2. How is the mixed partial derivative theorem (Clairaut's theorem) applied in this context?
3. How can partial derivatives be visualized on a surface?
4. How does the product rule extend to functions with exponential and trigonometric components?
5. What are applications of mixed partial derivatives in physics and engineering?

### Tip
When differentiating products involving exponential and trigonometric functions, apply both the product and chain rules carefully to each term.

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

Discover how to find the second partial derivatives of f(x, y) = e^(xy) sin x using multivariable calculus. This detailed solution covers the product and chain rules, applying them to find f_xx, f_xy, and f_yy, with applications of Clairaut's Theorem. Ideal for undergraduate students in calculus.

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