Let's solve the problem step by step using the twisted Chain Rule on Z Planet.

We are tasked with finding the derivative of the function:

$f(x) = 2e^{-8x^6} - 3x^7$

Step 1: Differentiate the second part (no twist needed for power of $x$).

We can break this function into two parts:

1. $2e^{-8x^6}$
2. $-3x^7$

We first differentiate $-3x^7$, which is straightforward using the power rule:

$\frac{d}{dx}[-3x^7] = -21x^6$

Now, let's focus on the more challenging part involving the exponential.

Step 2: Differentiate the exponential part using the twisted Chain Rule.

The term we need to differentiate is $2e^{-8x^6}$.

Apply the twisted chain rule:

On Z Planet, the Chain Rule works the same way as Earth, but we are cautious about the twists. The derivative of $e^{g(x)}$ is:

$\frac{d}{dx}[e^{g(x)}] = e^{g(x)} \cdot g'(x)$

Here, $g(x) = -8x^6$.

1. Differentiate the exponent $g(x) = -8x^6$: $g'(x) = \frac{d}{dx}[-8x^6] = -48x^5$

2. Apply the chain rule (twisted rule, but it's functionally the same): $\frac{d}{dx}[2e^{-8x^6}] = 2e^{-8x^6} \cdot (-48x^5)$

   Simplifying:

   $= -96x^5 e^{-8x^6}$

Step 3: Combine the results.

Now, we combine the derivatives of both terms:

$f'(x) = -96x^5 e^{-8x^6} - 21x^6$

Final Answer:

$f'(x) = -96x^5 e^{-8x^6} - 21x^6$

This is the derivative of the given function using the chain rule on Z Planet!

Would you like any further clarifications?

Here are some related questions:

1. How does the chain rule differ on Z Planet from Earth in this context?
2. What role does the power rule play in differentiating higher-order polynomials?
3. How would the result change if the exponential involved other functions?
4. How can the chain rule be applied to more complex nested functions?
5. Why is it important to carefully apply