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To differentiate the function \( m(t) = (8 - 3t)^{13} \), we can apply the **chain rule**.

Here’s the process:

1. Recognize that the function is a composition of two functions: the outer function \( f(u) = u^{13} \), and the inner function \( u(t) = 8 - 3t \).

2. Apply the chain rule: 
   \[
   \frac{d}{dt} \left[ (8 - 3t)^{13} \right] = 13(8 - 3t)^{12} \cdot \frac{d}{dt}(8 - 3t)
   \]

3. Now differentiate the inner function \( \frac{d}{dt}(8 - 3t) \):
   \[
   \frac{d}{dt}(8 - 3t) = -3
   \]

4. Multiply everything together:
   \[
   \frac{d}{dt} \left[ (8 - 3t)^{13} \right] = 13(8 - 3t)^{12} \cdot (-3)
   \]

5. Simplify:
   \[
   \frac{d}{dt} m(t) = -39(8 - 3t)^{12}
   \]

So the derivative is:
\[
m'(t) = -39(8 - 3t)^{12}
\]

Would you like more details or have any questions?

Here are 5 related questions for further practice:
1. What is the derivative of \( m(t) = (5 - 4t)^{10} \)?
2. How would you differentiate \( m(t) = (7t + 1)^{8} \)?
3. Can you apply the chain rule to \( f(x) = \sin(3x^2 + 2x) \)?
4. What is the second derivative of \( m(t) = (8 - 3t)^{13} \)?
5. How does the chain rule apply to trigonometric functions like \( \cos(2t) \)?

**Tip:** When differentiating powers of a linear function like \( (8 - 3t)^{13} \), always apply the chain rule to account for both the power and the inner function.

This detailed guide explains how to differentiate the function m(t) = (8 - 3t)^{13} using the chain rule. The solution provides step-by-step instructions and explores key calculus concepts, including the chain rule for composite functions. Suitable for students in Grades 11-12.

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