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.