Complete the table.

y = f(g(x)) u = g(x) y = f(u)
y = (2x − 4)^7 u = y=

This problem involves the chain rule of derivatives.

clearly

u(x) = 2x-4
f(u) = u^7
y = f(u) = u^7 = (2x-4)^7

Oh, the chain rule, what a thrill! Let's find those derivatives with a little skill!

First, let's find the derivative for u = g(x). To do this, we need to find the derivative of the function g(x). However, you didn't provide any information about what g(x) actually is. So, I'm afraid I can't complete the table without that missing piece.

But hey, don't fret, my friend! I'm here to put a smile on your face, even if we can't solve this case.

To solve this problem, we need to find the derivative of each function in the chain rule and then substitute them into the table.

First, let's find the derivative of g(x) and f(u):

1. Derivative of g(x):
Given the function u = g(x), we need to find du/dx (the derivative of u with respect to x). If g(x) is given as (2x - 4), then we can use the power rule to find its derivative. The power rule states that d/dx(x^n) = nx^(n-1), where n is a constant.

In this case, g(x) = (2x - 4), so using the power rule, we can find the derivative of g(x):
du/dx = d/dx(2x - 4) = 2 * 1 * x^(1-1) = 2

2. Derivative of f(u):
Given the function y = f(u), we need to find dy/du (the derivative of y with respect to u). The derivative of f(u) depends on the specific function f(u), which is not mentioned in the problem. Therefore, we cannot determine its derivative without additional information.

Now, we can fill in the table:

y = f(g(x)) | u = g(x) | y = f(u)
----------------|-----------------|----------------
y = (2x - 4)^7 | u = 2 | y = f(u)

In the given table, we have filled in the values for y = f(g(x)) and u = g(x) based on the given functions.

To find the value of u, we substitute g(x) = (2x - 4) into the equation u = g(x):
u = 2x - 4

To find the value of y, we substitute u = 2 into the equation y = f(u):
y = f(2)

However, since the specific function f(u) is not provided, we are unable to calculate the value of y = f(u) without further information.