Math Problem Statement

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).

Solution

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Math Problem Analysis

Mathematical Concepts

Calculus
Chain Rule
Derivatives
Exponential Functions

Formulas

Chain Rule: f(g(x))' = f'(g(x)) * g'(x)
Exponential function derivative: d/dx(e^g(x)) = e^g(x) * g'(x)

Theorems

Chain Rule Theorem
Derivative of Exponential Functions

Suitable Grade Level

Grades 11-12 and college level