To find the exact differential $du$ for the function $u = e^{2x} \sin(y)$, we need to compute the partial derivatives of $u$ with respect to $x$ and $y$ and apply the formula for the total differential:

$du = \frac{\partial u}{\partial x} dx + \frac{\partial u}{\partial y} dy$

Step 1: Compute the partial derivative of $u$ with respect to $x$.

$u = e^{2x} \sin(y)$

Since $\sin(y)$ is treated as a constant with respect to $x$, we apply the chain rule:

$\frac{\partial u}{\partial x} = \frac{\partial}{\partial x} \left( e^{2x} \right) \sin(y) = 2e^{2x} \sin(y)$

Step 2: Compute the partial derivative of $u$ with respect to $y$.

$u = e^{2x} \sin(y)$

Here, $e^{2x}$ is treated as a constant with respect to $y$, so:

$\frac{\partial u}{\partial y} = e^{2x} \cos(y)$

Step 3: Write the total differential $du$.

Now, combine the two partial derivatives:

$du = \left( 2e^{2x} \sin(y) \right) dx + \left( e^{2x} \cos(y) \right) dy$

Thus, the exact differential $du$ is:

$du = 2e^{2x} \sin(y) \, dx + e^{2x} \cos(y) \, dy$

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Would you like more details on how to apply this to specific problems, or do you have any questions? Here are a few related questions:

1. How do you apply the total differential in optimization problems?
2. What is the difference between an exact and an inexact differential?
3. Can the total differential be used to approximate the change in a function for small changes in $x$ and $y$?
4. How do you compute partial derivatives for functions involving products of exponential and trigonometric functions?
5. How would you find the total differential for a function of three variables?

Tip: When computing total differentials, always remember to apply the chain rule and treat other variables as constants when differentiating with respect to one variable.