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 (-3x^7 is an exponent with -8x^6)

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).

On Planet Z, the Chain Rule as we know it is still expressed as \( f(g(x))' = f'(g(x)) g'(x) \), which is consistent with Earth's mathematics. So we can apply the chain rule, recognizing that we need to adapt it carefully to the function given.

Let's solve the problem step by step.

The function is:

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

### Step 1: Identify the outer function and the inner function

- The **outer function** is \( f(g(x)) = 2e^{g(x)} \), where \( g(x) = -8x^6 - 3x^7 \).
- The **inner function** is \( g(x) = -8x^6 - 3x^7 \).

### Step 2: Apply the Chain Rule

Using the Chain Rule \( f(g(x))' = f'(g(x)) \cdot g'(x) \), we need to calculate both the derivative of the outer function \( f'(g(x)) \) and the derivative of the inner function \( g'(x) \).

#### a) Derivative of the outer function

The outer function is \( f(g(x)) = 2e^{g(x)} \). The derivative of \( e^{g(x)} \) is \( e^{g(x)} \), and since we have a constant multiple of 2, the derivative is:

\[
f'(g(x)) = 2e^{g(x)}
\]

#### b) Derivative of the inner function

The inner function is \( g(x) = -8x^6 - 3x^7 \). Taking the derivative of each term gives:

\[
g'(x) = -8 \cdot 6x^5 - 3 \cdot 7x^6 = -48x^5 - 21x^6
\]

### Step 3: Combine using the Chain Rule

Now, multiply the derivative of the outer function by the derivative of the inner function:

\[
f'(x) = 2e^{-8x^6 - 3x^7} \cdot (-48x^5 - 21x^6)
\]

This simplifies to:

\[
f'(x) = 2(-48x^5 - 21x^6)e^{-8x^6 - 3x^7}
\]

Or more neatly:

\[
f'(x) = (-96x^5 - 42x^6)e^{-8x^6 - 3x^7}
\]

### Final Answer:

\[
f'(x) = (-96x^5 - 42x^6)e^{-8x^6 - 3x^7}
\]

Would you like more details on any part of this solution?

### Expanding Questions:
1. How would this derivative change if the exponents in \( g(x) \) were different powers of \( x \)?
2. What are the implications of the constant factor \( 2 \) in the function on the final derivative?
3. Can the Chain Rule be used to differentiate functions that are compositions of more than two functions?
4. What if the exponent in the exponential was itself a more complex function of \( x \)?
5. How does the derivative of an exponential function behave compared to polynomial or trigonometric functions?

### Tip: 
Always identify the outer and inner functions first when applying the chain rule. This makes the process straightforward and minimizes mistakes.

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 (-3x^7 is an exponent with -8x^6) f'(x)=

Solve a challenging calculus problem involving the chain rule to find the derivative of the function 2e^(-8x^6 - 3x^7). This step-by-step solution explains how to apply the chain rule to composite functions, specifically exponential functions. Suitable for advanced high school students and college-level calculus learners.

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