To identify the inside and outside functions, we need to express the given function as a composition of two functions.

The given function is:

$y = (x^2 - 2x + 6)^5$

Step 1: Identify the inside function, $u = g(x)$

The expression inside the parentheses, $x^2 - 2x + 6$, is the inside function.

Thus:

$u = g(x) = x^2 - 2x + 6$

Step 2: Identify the outside function, $y = f(u)$

The outer part of the expression is the fifth power of the inside function. This is the outside function.

Thus:

$y = f(u) = u^5$

Final Answer:

• $u = g(x) = x^2 - 2x + 6$
• $y = f(u) = u^5$

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Would you like a detailed explanation of how to differentiate this function using the chain rule or any further questions on composition of functions? Here are a few related questions:

1. How do you apply the chain rule to differentiate a composite function?
2. What is the derivative of the function $y = (x^2 - 2x + 6)^5$?
3. Can you identify the inside and outside functions for the expression $y = \sin(x^2 + 3x)$?
4. How do you find the derivative of a composition of functions with multiple terms inside?
5. What is the importance of recognizing inside and outside functions in calculus?

Tip: The chain rule is essential when differentiating composite functions—always identify the inside and outside functions first!