To find the derivative of $f(x) = (7x^6 - 6x)^7$, we will use the chain rule, which states that if $g(x) = (h(x))^n$, then:

$g'(x) = n \cdot (h(x))^{n-1} \cdot h'(x)$

Here, let $h(x) = 7x^6 - 6x$ and $n = 7$. Therefore, the derivative $f'(x)$ will be:

$f'(x) = 7 \cdot (7x^6 - 6x)^6 \cdot \frac{d}{dx} (7x^6 - 6x)$

Now, we need to differentiate $7x^6 - 6x$ with respect to $x$:

$\frac{d}{dx} (7x^6 - 6x) = 42x^5 - 6$

Now, putting everything together:

$f'(x) = 7 \cdot (7x^6 - 6x)^6 \cdot (42x^5 - 6)$

Thus, the derivative of $f(x) = (7x^6 - 6x)^7$ is:

$f'(x) = 7(7x^6 - 6x)^6 \cdot (42x^5 - 6)$

Let me know if you need further clarification or details.

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Here are 5 related questions you might find helpful:

1. What is the chain rule, and how does it apply to differentiating composite functions?
2. How do you differentiate polynomials like $7x^6$ or $6x$?
3. What are higher-order derivatives, and how do we find them for composite functions?
4. How can you simplify the expression for $f'(x)$?
5. What would be the second derivative of $f(x)$?

Tip: Remember to always check if the function requires the chain rule by identifying a composition of functions (like $(g(x))^n$).