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.