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The given function is:

\[
w = \left( t^7 + 1 \right)^{30}
\]

To find the derivative \( w' \) with respect to \( t \), we'll use the **chain rule**.

### Step-by-step process:

1. **Outer Function:** The outer function is \( \left( t^7 + 1 \right)^{30} \), so we differentiate it as we would for \( u^{30} \) where \( u = t^7 + 1 \).
   \[
   \frac{d}{dt}\left( u^{30} \right) = 30u^{29} \cdot \frac{du}{dt}
   \]

2. **Inner Function:** The inner function is \( u = t^7 + 1 \), and its derivative is:
   \[
   \frac{d}{dt}\left( t^7 + 1 \right) = 7t^6
   \]

### Combining the results:

Using the chain rule:

\[
w' = 30 \cdot \left( t^7 + 1 \right)^{29} \cdot 7t^6
\]

Simplifying:

\[
w' = 210t^6 \cdot \left( t^7 + 1 \right)^{29}
\]

Thus, the derivative of the given function is:

\[
w' = 210t^6 \left( t^7 + 1 \right)^{29}
\]

Would you like any additional details or have any questions?

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Here are 5 related questions to practice:
1. What is the derivative of \( \left( 2t^4 + 3 \right)^{15} \)?
2. How does the chain rule apply to trigonometric functions?
3. Find the second derivative of \( w = \left( t^7 + 1 \right)^{30} \).
4. How do we handle functions that involve a product of terms like \( (t^7+1)^2(t^3+5) \)?
5. What is the derivative of \( \left( t^8 + 2t \right)^5 \)?

**Tip:** Always break down complicated functions into outer and inner components to apply the chain rule efficiently.

Find the derivative of the function w = (t^7 + 1)^30.

Learn how to differentiate the function w = (t^7 + 1)^30 with respect to t using the chain rule. This step-by-step solution explains how to handle outer and inner functions for students in Grades 11-12.

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