In 2040, people all over the world were excited to discover Planet Z populated by friendly aliens. Scientists were working hard to make it technically possible to meet the new distant and long-awaited neighbors. This was difficult as many laws of nature on the planet appeared to be… “shifted“ compared to our blue planet. Additionally, most of the software/online problem solvers couldn’t help in preparations since they are tuned to our home planet environment. Even worse, people of 2040s rely on the computers completely and can do almost nothing “by hand” but now you're on board the spaceship #159881477, 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^−8x^6−3x^7

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).

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!

In 2040, people all over the world were excited to discover Planet Z populated by friendly aliens. Scientists were working hard to make it technically possible to meet the new distant and long-awaited neighbors. This was difficult as many laws of nature on the planet appeared to be… 'shifted' compared to our blue planet. Additionally, most of the software/online problem solvers couldn’t help in preparations since they are tuned to our home planet environment. Even worse, people of 2040s rely on the computers completely and can do almost nothing 'by hand' but now you're on board the spaceship #159881477, 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^−8x^6−3x^7.

Learn how to apply the chain rule to find the derivative of f(x) = 2e^{-8x^6} - 3x^7. This calculus problem demonstrates the application of key concepts like the chain rule and exponential differentiation. Ideal for undergraduate students learning advanced calculus.

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