 using the first derivative property of laplace transform, find f'(t)=3e^{2t} + 4\sin(3t)

To solve for \( f'(t) = 3e^{2t} + 4\sin(3t) \) using the first derivative property of the Laplace transform, we will proceed step by step.

### Step 1: Recall the first derivative property of the Laplace transform

The first derivative property states that:

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

where \( F(s) \) is the Laplace transform of \( f(t) \), and \( f(0) \) is the initial condition of \( f(t) \) at \( t = 0 \).

### Step 2: Express \( f'(t) \)

Given:
\[
f'(t) = 3e^{2t} + 4\sin(3t)
\]

We need to take the Laplace transform of each term.

#### (a) Laplace Transform of \( 3e^{2t} \)

Using the Laplace transform of \( e^{at} \), which is \( \mathcal{L} \{ e^{at} \} = \frac{1}{s - a} \), we get:

\[
\mathcal{L} \{ 3e^{2t} \} = 3 \cdot \frac{1}{s - 2} = \frac{3}{s - 2}
\]

#### (b) Laplace Transform of \( 4\sin(3t) \)

Using the Laplace transform of \( \sin(at) \), which is \( \mathcal{L} \{ \sin(at) \} = \frac{a}{s^2 + a^2} \), we get:

\[
\mathcal{L} \{ 4\sin(3t) \} = 4 \cdot \frac{3}{s^2 + 9} = \frac{12}{s^2 + 9}
\]

### Step 3: Add the Laplace transforms

Now, the Laplace transform of \( f'(t) \) is the sum of the transforms:

\[
\mathcal{L} \{ f'(t) \} = \frac{3}{s - 2} + \frac{12}{s^2 + 9}
\]

### Step 4: Use the first derivative property to find \( \mathcal{L} \{ f(t) \} \)

Using the property \( \mathcal{L} \{ f'(t) \} = sF(s) - f(0) \), and assuming \( f(0) = 0 \) (if no initial condition is given), we have:

\[
sF(s) = \frac{3}{s - 2} + \frac{12}{s^2 + 9}
\]

Thus:

\[
F(s) = \frac{\frac{3}{s - 2} + \frac{12}{s^2 + 9}}{s}
\]

This expression represents the Laplace transform of \( f(t) \).

### Conclusion

Using the first derivative property of the Laplace transform, we found that the Laplace transform of the original function \( f(t) \) is:

\[
F(s) = \frac{3}{s(s - 2)} + \frac{12}{s(s^2 + 9)}
\]

If you'd like to go further and invert the transform to find \( f(t) \), or need more details on any step, let me know!

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Do you want to see the details of the inverse transform or have any specific questions?

Here are some related problems you could explore:
1. Find the inverse Laplace transform of \( \frac{3}{s(s - 2)} \).
2. Use the Laplace transform to solve \( f''(t) = 3e^{2t} + 4\sin(3t) \) with \( f(0) = 0 \), \( f'(0) = 1 \).
3. Solve a second-order differential equation using Laplace transforms.
4. Explore the Laplace transform of \( e^{at} \cos(bt) \).
5. How does the second derivative property of the Laplace transform work?

**Tip:** The Laplace transform is particularly useful for solving differential equations with initial conditions, as it simplifies them into algebraic equations.

Using the first derivative property of Laplace transform, find f'(t)=3e^{2t} + 4\sin(3t).

Explore how to solve the differential equation f'(t) = 3e^{2t} + 4sin(3t) using the first derivative property of the Laplace transform. Step-by-step breakdown including key formulas and transformations, suitable for advanced high school or undergraduate students.

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