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Solve all the questions correctly with clear writing, showing the logical steps. 

To solve the question provided in the image, let's carefully find \(\frac{dy}{dt}\) for \(y = e^{\sin^2(\pi t - 11)}\).

### Steps:

1. **Given Function**:
   \[
   y = e^{\sin^2(\pi t - 11)}
   \]

2. **Chain Rule**:
   To differentiate \(y\) with respect to \(t\), use the chain rule:
   \[
   \frac{dy}{dt} = \frac{d}{dt}\left(e^{\sin^2(\pi t - 11)}\right).
   \]

3. **Exponential Rule**:
   The derivative of \(e^u\) is \(e^u \cdot \frac{du}{dt}\). Here, \(u = \sin^2(\pi t - 11)\). Applying this rule:
   \[
   \frac{dy}{dt} = e^{\sin^2(\pi t - 11)} \cdot \frac{d}{dt}\left(\sin^2(\pi t - 11)\right).
   \]

4. **Differentiate \(\sin^2(\pi t - 11)\)**:
   Use the chain rule again. The derivative of \(\sin^2(u)\) is \(2\sin(u)\cos(u) \cdot \frac{du}{dt}\), where \(u = \pi t - 11\). Hence:
   \[
   \frac{d}{dt} \sin^2(\pi t - 11) = 2\sin(\pi t - 11)\cos(\pi t - 11) \cdot \frac{d}{dt}(\pi t - 11).
   \]

5. **Differentiate \(\pi t - 11\)**:
   The derivative of \(\pi t - 11\) with respect to \(t\) is simply \(\pi\).

6. **Combine the Results**:
   Substitute back into the expression for \(\frac{dy}{dt}\):
   \[
   \frac{dy}{dt} = e^{\sin^2(\pi t - 11)} \cdot \left(2\sin(\pi t - 11)\cos(\pi t - 11) \cdot \pi\right).
   \]

7. **Simplify**:
   Using the double-angle identity, \(2\sin(u)\cos(u) = \sin(2u)\), the expression simplifies to:
   \[
   \frac{dy}{dt} = e^{\sin^2(\pi t - 11)} \cdot \pi \sin(2(\pi t - 11)).
   \]

### Final Answer:
\[
\frac{dy}{dt} = \pi e^{\sin^2(\pi t - 11)} \sin(2(\pi t - 11)).
\]

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Would you like further clarification or a step-by-step breakdown of a specific part?  

### Related Questions:
1. How is the chain rule applied when differentiating exponential functions with trigonometric components?
2. What is the significance of the double-angle identity in simplifying derivatives?
3. Can you provide an example where similar exponential and trigonometric functions arise in physics or engineering?
4. What happens if the argument of the sine function includes additional constants or variables?
5. How would you compute the second derivative (\(\frac{d^2y}{dt^2}\)) of this function?

### Tip:
Always simplify intermediate steps where possible to make the final derivative easier to interpret and apply.

Solve all the questions correctly with clear writing, showing the logical steps.

Step-by-step solution to find dy/dt for y = e^(sin^2(pi t - 11)) using chain rule, exponential differentiation, and trigonometric identities. This problem involves calculus concepts from exponential and trigonometric functions. Suitable for Calculus I students.

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