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To derive the given function \( T(t) = -0.009t^3 + 0.29t^2 - 1.7t + 15.5 \), we will differentiate each term with respect to \( t \).

### Steps to differentiate:

1. **Differentiate \( -0.009t^3 \):**
   \[
   \frac{d}{dt}(-0.009t^3) = -0.009 \cdot 3t^2 = -0.027t^2
   \]

2. **Differentiate \( 0.29t^2 \):**
   \[
   \frac{d}{dt}(0.29t^2) = 0.29 \cdot 2t = 0.58t
   \]

3. **Differentiate \( -1.7t \):**
   \[
   \frac{d}{dt}(-1.7t) = -1.7
   \]

4. **Differentiate \( 15.5 \):**
   \[
   \frac{d}{dt}(15.5) = 0
   \]

### Therefore, the derivative is:
\[
T'(t) = -0.027t^2 + 0.58t - 1.7
\]

Would you like more details on any part of the process?

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Here are five related questions:
1. How do you differentiate polynomials in general?
2. What are the rules of differentiation (e.g., power rule, constant rule)?
3. What does the derivative of a function tell you about the function's graph?
4. Can you explain how to find the second derivative of a function?
5. How do derivatives apply in real-world scenarios like physics?

**Tip:** Always remember to apply the power rule when differentiating terms involving powers of \( t \).

Derive T(t) = -0.009t^3 + 0.29t^2 - 1.7t + 15.5

Learn how to differentiate the polynomial function T(t) = -0.009t^3 + 0.29t^2 - 1.7t + 15.5 using the power rule and constant rule. The step-by-step process is explained clearly and is suitable for students in Grades 10-12.

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