The captain is asking for a derivative using the "twisted" version of the Chain Rule from Planet Z. Based on what you’ve mentioned, the only difference on Z Planet is that the Chain Rule remains the same as Earth's, so we can safely use it here. Let's find the derivative of the function:

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

We need to differentiate each term separately:

Step 1: Differentiate $2e^{-8x^6}$

• Treat the function as a composition of two functions: $g(x) = -8x^6$ and $h(g(x)) = e^{g(x)}$.
• Apply the Chain Rule: $\frac{d}{dx} \left( 2e^{-8x^6} \right) = 2 \cdot \frac{d}{dg} \left( e^g \right) \cdot \frac{dg}{dx}$ Since $\frac{d}{dg} e^g = e^g$, and $g(x) = -8x^6$, we now find $\frac{dg}{dx}$: $\frac{d}{dx} (-8x^6) = -48x^5$ Thus, the derivative of the first term is: $2 \cdot e^{-8x^6} \cdot (-48x^5) = -96x^5 e^{-8x^6}$

Step 2: Differentiate $-3x^7$

This is a straightforward power rule differentiation: $\frac{d}{dx} \left( -3x^7 \right) = -21x^6$

Step 3: Combine the results

Now, combining the derivatives of both terms: $f'(x) = -96x^5 e^{-8x^6} - 21x^6$

This is the derivative of $f(x) = 2e^{-8x^6} - 3x^7$ using the twisted Chain Rule (which, luckily for us, is unchanged on Planet Z).

Let me know if you'd like further details or explanations!

5 Related Questions:

1. How does the Chain Rule apply to multi-variable functions on Planet Z?
2. What if the exponent in $e^{-8x^6}$ was a more complex function of $x$?
3. How do changes in physical laws on Planet Z affect mathematical computations like integration?
4. Can you solve an integral of $f(x)$ on Planet Z using similar rules?
5. How do higher-order derivatives differ on Planet Z from Earth?

Tip:

When applying the Chain Rule, always identify the inner and outer functions clearly. This makes applying the rule systematically easier!