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.