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To find the derivative of the function \( F(t) = e^{6t \sin(2t)} \), we need to use the chain rule. Here’s the step-by-step solution:

### Step-by-Step Solution

1. **Identify the outer and inner functions**:
   - The outer function is \( e^u \) where \( u = 6t \sin(2t) \).
   - The inner function is \( u = 6t \sin(2t) \).

2. **Differentiate the outer function** \( e^u \) with respect to \( u \):
   \[
   \frac{d}{du} (e^u) = e^u
   \]

3. **Differentiate the inner function** \( 6t \sin(2t) \) with respect to \( t \) using the product rule:
   \[
   u' = \frac{d}{dt} (6t \sin(2t)) = 6 \sin(2t) + 6t \cdot \frac{d}{dt} (\sin(2t))
   \]
   - Here, \(\frac{d}{dt} (\sin(2t)) = 2 \cos(2t)\) (using the chain rule again).
   \[
   u' = 6 \sin(2t) + 6t \cdot 2 \cos(2t) = 6 \sin(2t) + 12t \cos(2t)
   \]

4. **Combine using the chain rule**:
   - Now we apply the chain rule: \( F'(t) = e^{6t \sin(2t)} \cdot u' \).
   \[
   F'(t) = e^{6t \sin(2t)} \cdot (6 \sin(2t) + 12t \cos(2t))
   \]

Thus, the derivative of the function is:
\[
F'(t) = e^{6t \sin(2t)} \left(6 \sin(2t) + 12t \cos(2t)\right)
\]

Would you like more details on this solution or have any questions?

### Related Questions:
1. How do you apply the product rule for derivatives?
2. What are the steps to apply the chain rule to composite functions?
3. How does the derivative of \( \sin(kt) \) differ from that of \( \sin(t) \)?
4. What is the significance of using the chain rule in differentiation?
5. Can you show an example of differentiating a function with a nested exponent?

### Tip:
When differentiating exponential functions with a complex exponent, breaking it down using the chain rule makes it much easier to handle.

Find the derivative of the function F(t) = e^{6t \sin(2t)}.

This page covers how to find the derivative of the exponential function F(t) = e^{6t \sin(2t)}. It includes step-by-step application of the chain and product rules in calculus, suitable for high school seniors or early university students.

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