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The function provided is:

\[
f(z) = \left(e^z + 6 \sin(z)\right)^3
\]

We are tasked with finding the derivative \(\frac{df}{dz}\) using the power rule and expressing the result in exact form.

### Steps to solve:

1. **Outer function**: Apply the power rule to the outermost part of the function \(\left(e^z + 6 \sin(z)\right)^3\). 
   - The power rule states \(\frac{d}{dz} \left( g(z)^n \right) = n \cdot g(z)^{n-1} \cdot g'(z)\).
   - Here \(n = 3\) and \(g(z) = e^z + 6 \sin(z)\).

   So, applying the power rule:

   \[
   \frac{df}{dz} = 3 \left(e^z + 6 \sin(z)\right)^2 \cdot \frac{d}{dz}\left(e^z + 6 \sin(z)\right)
   \]

2. **Inner derivative**: Now, find the derivative of the inner function \(e^z + 6 \sin(z)\).
   - The derivative of \(e^z\) is \(e^z\).
   - The derivative of \(6 \sin(z)\) is \(6 \cos(z)\).

   So:

   \[
   \frac{d}{dz}\left(e^z + 6 \sin(z)\right) = e^z + 6 \cos(z)
   \]

3. **Final expression**: Substituting the inner derivative back into the equation:

   \[
   \frac{df}{dz} = 3 \left(e^z + 6 \sin(z)\right)^2 \cdot \left(e^z + 6 \cos(z)\right)
   \]

Thus, the derivative of the function \(f(z)\) is:

\[
\frac{df}{dz} = 3 \left(e^z + 6 \sin(z)\right)^2 \left(e^z + 6 \cos(z)\right)
\]

---

Would you like further details or have any questions?

Here are 5 related questions to explore this further:
1. How does the chain rule work in more complex compositions?
2. Can you apply the same process to a different power, such as \( (e^z + 6 \sin(z))^5 \)?
3. What is the derivative of a product of two functions, like \( e^z \cdot \sin(z) \)?
4. How would you differentiate \( e^z + 6 \sin(z) \) if it were raised to a different exponent, such as \( \frac{3}{2} \)?
5. What are the potential real-world applications of such derivatives in physics or engineering?

**Tip:** When using the chain rule, always remember to differentiate the outer function first, leaving the inner function unchanged, and then multiply by the derivative of the inner function.

Find the derivative of the function using the Power Rule: f(z) = (e^z + 6sin(z))^3. Express numbers in exact form. Use symbolic notation and fractions where needed.

This page explains how to find the derivative of f(z) = (e^z + 6sin(z))^3 using the Power Rule and Chain Rule. Learn step-by-step with the exact form of the derivative and detailed breakdown of the process.

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