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.