To find the derivative of the given function:

$f(x) = \sqrt[8]{\left(6x^7 + x^2 - 1\right)^9}$

We can rewrite this using exponent rules as:

$f(x) = \left(6x^7 + x^2 - 1\right)^{\frac{9}{8}}.$

Now, apply the chain rule to differentiate. The derivative of $f(x)$ is:

$f'(x) = \frac{9}{8} \left(6x^7 + x^2 - 1\right)^{\frac{9}{8} - 1} \cdot \frac{d}{dx}\left(6x^7 + x^2 - 1\right).$

Step 1: Differentiate the inner function $6x^7 + x^2 - 1$:

$\frac{d}{dx}(6x^7 + x^2 - 1) = 42x^6 + 2x.$

Step 2: Substitute into the chain rule:

$f'(x) = \frac{9}{8} \left(6x^7 + x^2 - 1\right)^{\frac{1}{8}} \cdot (42x^6 + 2x).$

Final Answer:

$f'(x) = \frac{9}{8} (6x^7 + x^2 - 1)^{\frac{1}{8}} \cdot (42x^6 + 2x).$

Let me know if you'd like further clarification or a step-by-step breakdown!

Here are 5 related questions to explore further:

1. How would you compute the derivative of a general power function using the chain rule?
2. Can you simplify this derivative for $x = 0$ or any specific values of $x$?
3. How does the chain rule apply when there are multiple nested functions?
4. What is the significance of rewriting the function in exponential form before differentiation?
5. How would this change if the original function had a higher or lower root?

Tip: Always simplify the function as much as possible before applying differentiation rules to make calculations easier.