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−8x6−3x7f(x)=2e-8x6-3x7

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).

Let's solve the problem step by step using the **twisted Chain Rule** on Z Planet.

We are tasked with finding the derivative of the function:

\[
f(x) = 2e^{-8x^6} - 3x^7
\]

### Step 1: Differentiate the second part (no twist needed for power of \( x \)).

We can break this function into two parts:

1. \( 2e^{-8x^6} \)
2. \( -3x^7 \)

We first differentiate \( -3x^7 \), which is straightforward using the power rule:

\[
\frac{d}{dx}[-3x^7] = -21x^6
\]

Now, let's focus on the more challenging part involving the exponential.

### Step 2: Differentiate the exponential part using the twisted Chain Rule.

The term we need to differentiate is \( 2e^{-8x^6} \).

#### Apply the twisted chain rule:

On Z Planet, the Chain Rule works the same way as Earth, but we are cautious about the twists. The derivative of \( e^{g(x)} \) is:

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

Here, \( g(x) = -8x^6 \).

1. **Differentiate the exponent \( g(x) = -8x^6 \):**
   \[
   g'(x) = \frac{d}{dx}[-8x^6] = -48x^5
   \]

2. **Apply the chain rule (twisted rule, but it's functionally the same):**
   \[
   \frac{d}{dx}[2e^{-8x^6}] = 2e^{-8x^6} \cdot (-48x^5)
   \]

   Simplifying:

   \[
   = -96x^5 e^{-8x^6}
   \]

### Step 3: Combine the results.

Now, we combine the derivatives of both terms:

\[
f'(x) = -96x^5 e^{-8x^6} - 21x^6
\]

### Final Answer:

\[
f'(x) = -96x^5 e^{-8x^6} - 21x^6
\]

This is the derivative of the given function using the chain rule on Z Planet!

Would you like any further clarifications?

Here are some related questions:
1. How does the chain rule differ on Z Planet from Earth in this context?
2. What role does the power rule play in differentiating higher-order polynomials?
3. How would the result change if the exponential involved other functions?
4. How can the chain rule be applied to more complex nested functions?
5. Why is it important to carefully apply

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−8x6−3x7

Solve the derivative of the function 2e^(-8x^6) - 3x^7 using the Chain Rule on Planet Z. Learn about the application of the Chain Rule, Exponential Functions, and Polynomial Differentiation in this detailed step-by-step solution. Suitable for undergraduate calculus students.

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