To compute the Laplace transform of $f(t) = \frac{e^{3t}}{\sin(2t)}$ using the first derivative property, let's follow the steps.

Step-by-Step Procedure:

1. Identify the First Derivative Property:

The first derivative property of the Laplace transform states:

$\mathcal{L}\{f'(t)\} = sF(s) - f(0)$ where $F(s) = \mathcal{L}\{f(t)\}$.

2. Rewrite the Function:

We have $f(t) = \frac{e^{3t}}{\sin(2t)}$, and we are tasked with finding $F(s) = \mathcal{L}\left\{\frac{e^{3t}}{\sin(2t)}\right\}$.

Let's define $f(t)$ as $f(t) = g(t)e^{3t}$, where $g(t) = \frac{1}{\sin(2t)}$.

3. Differentiate $f(t)$:

Apply the product rule to differentiate $f(t) = g(t)e^{3t}$:

$f'(t) = g'(t)e^{3t} + 3g(t)e^{3t}$

Since $g(t) = \frac{1}{\sin(2t)}$, differentiate this with respect to $t$:

$g'(t) = -\frac{2\cos(2t)}{\sin^2(2t)}$

Therefore:

$f'(t) = \left( -\frac{2\cos(2t)}{\sin^2(2t)} \right) e^{3t} + 3\frac{e^{3t}}{\sin(2t)}$

4. Laplace Transform of $f'(t)$:

Using the derivative property of the Laplace transform:

$\mathcal{L}\{f'(t)\} = sF(s) - f(0)$

We already know $f(t) = \frac{e^{3t}}{\sin(2t)}$, so $f(0) = \frac{1}{\sin(0)}$, but $\sin(0) = 0$, and thus $f(0)$ is undefined. Therefore, this approach leads to complications, and the derivative property can't be directly applied to this function due to the singularity at $t = 0$.

Conclusion:

The function $f(t) = \frac{e^{3t}}{\sin(2t)}$ has a singularity at $t = 0$, and applying the first derivative property of the Laplace transform involves complexities due to this singularity. The Laplace transform of such functions generally requires the use of advanced techniques like partial fraction decomposition or contour integration, or handling singularities through special methods.

Would you like more detailed explanations of alternative methods to solve this, or do you have other questions?

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Here are some related questions you might find helpful:

1. How does one handle Laplace transforms of functions with singularities?
2. What are the conditions for applying the first derivative property of Laplace transforms?
3. How do you compute Laplace transforms using contour integration?
4. Can Laplace transforms be computed for all types of functions?
5. What is the relationship between Laplace transforms and Fourier transforms?

Tip: Always check the behavior of a function at $t = 0$ when using properties of the Laplace transform, as singularities can complicate direct applications.