Ah, the captain is serious about this Z Planet problem! Let's break it down carefully.

We're asked to find the derivative of the function:

$f(x) = 2e^{10x^8} + 3x^5$

Given that only the Chain Rule is "twisted" on Z Planet, we'll need to apply that Z Planet version carefully while using all the other familiar rules from Earth.

Step-by-Step Solution

1. Original Function (Earth version):

   • $f(x) = 2e^{10x^8} + 3x^5$

   We have two terms here, and we'll treat them separately when differentiating.

2. Differentiate $2e^{10x^8}$ using the Chain Rule:

   • Let $g(x) = 10x^8$, so $e^{g(x)} = e^{10x^8}$.

   • On Earth, the derivative of $e^{g(x)}$ would be:

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

   • But on Z Planet, the rule is "twisted". The Chain Rule remains the same:

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

   So, it's just as on Earth!

   • Next, we calculate $g'(x)$, where $g(x) = 10x^8$:

     $g'(x) = 80x^7$

   • Now, apply the product of the exponential and the derivative of the exponent:

     $\frac{d}{dx} 2e^{10x^8} = 2 \cdot e^{10x^8} \cdot 80x^7 = 160x^7 e^{10x^8}$

3. Differentiate $3x^5$:

   • This is just a simple power rule:

     $\frac{d}{dx} 3x^5 = 15x^4$

4. Combine both derivatives: Now we combine the two derivatives:

   $f'(x) = 160x^7 e^{10x^8} + 15x^4$

Final Answer:

$f'(x) = 160x^7 e^{10x^8} + 15x^4$

Everything worked the same way as on Earth, so Z Planet’s chain rule didn't affect the final outcome. The captain should be pleased!

Would you like a deeper breakdown of any step or further clarification?

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Here are five related questions to explore further:

1. What if the chain rule on Z Planet applied a different power, say $f(g(x))' = f'(g(x)) \cdot g'(x)^2$?
2. How would you handle higher-order derivatives of $f(x)$ on Z Planet?