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The given function is:

\[
y = 9t^2 + 2e^t
\]

To find the derivative \( y' \), we will differentiate each term of the function with respect to \( t \).

1. The derivative of \( 9t^2 \) is \( 18t \), using the power rule: \( \frac{d}{dt}(t^n) = nt^{n-1} \).
2. The derivative of \( 2e^t \) is \( 2e^t \), because \( e^t \) remains \( e^t \) when differentiated.

Thus, the derivative \( y' \) is:

\[
y' = 18t + 2e^t
\]

Would you like a further breakdown or details on this solution?

Here are 5 related questions to further your understanding:
1. What is the derivative of \( e^{kt} \) for any constant \( k \)?
2. How do you apply the power rule when differentiating a function like \( t^n \)?
3. Can you differentiate \( y = t^2 \sin(t) + 3e^t \)?
4. How would you find the second derivative of \( y = 9t^2 + 2e^t \)?
5. What is the derivative of a product of functions, like \( t^2 e^t \)?

**Tip:** Always apply the power rule and chain rule carefully when differentiating complex functions involving powers and exponentials.

Find the derivative of the function y = 9t^2 + 2e^t.

Learn how to find the derivative of the function y = 9t^2 + 2e^t using the power rule and exponential derivative rule. Suitable for calculus students in Grades 11-12 or higher.

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