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To differentiate the given function with respect to \( t \), let's start by breaking down the equation:

\[
y = a \cos(t) + t^2 \sin(t)
\]

### Step-by-step differentiation:

1. **Differentiate \( a \cos(t) \) with respect to \( t \):**
   Since \( a \) is a constant, we use the derivative of \( \cos(t) \), which is \( -\sin(t) \):
   \[
   \frac{d}{dt} \left( a \cos(t) \right) = -a \sin(t)
   \]

2. **Differentiate \( t^2 \sin(t) \) with respect to \( t \):**
   This is a product of two functions, \( t^2 \) and \( \sin(t) \). We will apply the product rule:
   \[
   \frac{d}{dt} \left( t^2 \sin(t) \right) = \frac{d}{dt}(t^2) \cdot \sin(t) + t^2 \cdot \frac{d}{dt}(\sin(t))
   \]
   - The derivative of \( t^2 \) is \( 2t \).
   - The derivative of \( \sin(t) \) is \( \cos(t) \).
   So:
   \[
   \frac{d}{dt} \left( t^2 \sin(t) \right) = 2t \sin(t) + t^2 \cos(t)
   \]

### Putting it all together:
\[
\frac{dy}{dt} = -a \sin(t) + \left( 2t \sin(t) + t^2 \cos(t) \right)
\]

Thus, the derivative of \( y \) with respect to \( t \) is:
\[
\boxed{\frac{dy}{dt} = -a \sin(t) + 2t \sin(t) + t^2 \cos(t)}
\]

Would you like a more detailed explanation of any part of the process? 

### Relative Questions:
1. What is the product rule in differentiation and when is it used?
2. Can you explain why the derivative of \( \cos(t) \) is \( -\sin(t) \)?
3. How would the answer change if \( a \) were a function of \( t \) instead of a constant?
4. What happens if we differentiate \( t^2 \sin(t) \) multiple times?
5. How do you apply the chain rule if there is a composite function like \( \cos(2t) \) in place of \( \cos(t) \)?

### Tip:
Always remember to apply the product rule when differentiating products of functions like \( t^2 \sin(t) \), and ensure constants are handled properly in differentiation.

Differentiate with respect to t: y = a cos(t) + t^2 sin(t)

Learn how to differentiate y = a cos(t) + t^2 sin(t) with respect to t using the product rule. This step-by-step solution covers key calculus concepts including the derivative of trigonometric functions and the product rule. Ideal for Grades 11-12 or college-level calculus students.

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