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

f'(x)= 

!!! 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).

It looks like we're aboard spaceship #1725486563, solving the "Z Planet" problem where the Chain Rule is modified. On Earth, the Chain Rule is typically:

\[
\frac{d}{dx} \left( f(g(x)) \right) = f'(g(x)) \cdot g'(x)
\]

However, on **Z Planet**, it's twisted:

\[
\frac{d}{dx} \left( f(g(x)) \right) = \frac{f'(g(x))}{g'(x)}
\]

### Given function: 
\[
f(x) = 2e^{10x^8 + 3x^5}
\]

Let's break it down.

---

### Step 1: Identify the outer and inner functions.

The given function $ f(x) $ is of the form $ f(x) = 2e^{g(x)} $, where:

\[
g(x) = 10x^8 + 3x^5
\]

### Step 2: Earth version of the derivative

First, calculate the derivative using Earth’s Chain Rule:

\[
f'(x) = 2 \cdot \frac{d}{dx} \left( e^{g(x)} \right)
\]

The derivative of $ e^{g(x)} $ with respect to $ x $ is:

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

Now, compute $ g'(x) $:

\[
g'(x) = \frac{d}{dx} \left( 10x^8 + 3x^5 \right) = 80x^7 + 15x^4
\]

Thus, the Earth version of the derivative is:

\[
f'(x) = 2 \cdot e^{10x^8 + 3x^5} \cdot (80x^7 + 15x^4)
\]

### Step 3: Z Planet version of the derivative

According to Z Planet’s twisted Chain Rule, we divide by $ g'(x) $ instead of multiplying by it. Thus, on Z Planet:

\[
f'(x) = \frac{2 \cdot e^{10x^8 + 3x^5}}{80x^7 + 15x^4}
\]

This is the derivative on Z Planet.

---

### Final answer:
\[
f'(x) = \frac{2e^{10x^8 + 3x^5}}{80x^7 + 15x^4}
\]

---

Would you like more details or have any questions?

### 5 Related Questions:
1. How would the derivative change if the exponent was a polynomial of higher degree?
2. What is the significance of the Chain Rule in calculus?
3. How would the Earth Chain Rule handle nested functions (e.g., $ e^{e^{g(x)}} $)?
4. What happens if you take the second derivative using Z Planet's rules?
5. How does the Z Planet rule affect implicit differentiation?

### Tip:
Always double-check if the inner function's derivative is computed correctly, especially on Z Planet, where the rules twist familiar patterns.

Use the chain rule to find the derivative of f(x) = 2e^{10x^8+3x^5}. The Chain Rule on Z Planet is twisted as f(g(x))' = f'(g)/g'(x).

Explore the unique 'Z Planet' Chain Rule to differentiate the function f(x) = 2e^{10x^8 + 3x^5}. The Chain Rule is modified, requiring division by the inner function's derivative. Detailed steps and solution provided for undergraduate calculus students.

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