but now you're on board the spaceship #1725486563, and the captain is asking to solve "ZP" (Z Planet) problem (you know what it means when captain is "asking"... that's an order):

Use the chain rule to find the derivative of f(x)=2e10x8+3x5f(x)=2e10x8+3x5

f'(x)=f′(x)=  

!!! The challenge is that the Chain Rule on Earth, f(g(x))'=f'(g)g'(x)f(g(x))′=f′(g)g′(x) ,  is "twisted" on Z planet as the following:

 f(g(x))'=f'(g)g'(x)f(g(x))′=f′(g)g′(x)    (all the other rules have not been changed).

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?

---

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?
3.

But now you're on board the spaceship #1725486563, and the captain is asking to solve 'ZP' (Z Planet) problem (you know what it means when captain is 'asking'... that's an order): Use the chain rule to find the derivative of f(x)=2e10x8+3x5. The challenge is that the Chain Rule on Earth, f(g(x))'=f'(g)g'(x), is 'twisted' on Z planet as the following: f(g(x))'=f'(g)g'(x) (all the other rules have not been changed).

Explore how to solve the derivative problem for f(x) = 2e^10x^8 + 3x^5 using the Chain Rule on Z Planet. This problem involves key calculus concepts like differentiation, exponential functions, and the Chain Rule, with a detailed step-by-step solution. Suitable for university-level students.

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