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

$f_{xx} = e^{xy}(-\sin x + y \cos x) + y e^{xy} \cos x + y^2 e^{xy} \sin x$ $f_{xy} = e^{xy}(x \cos x + \sin x + xy \sin x)$ $f_{yy} = x^2 e^{xy} \sin x$

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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.