You're right to point out the confusion. Let me clarify the steps.

When differentiating a composite function like \( f(x) = (x - 2)^5 \), we identify the inner and outer functions to apply the chain rule correctly.

### Chain Rule Refresher

The **chain rule** states that if you have a function that is a composition of two functions, say \( f(x) = g(u(x)) \), then the derivative of \( f(x) \) with respect to \( x \) is:
\[
f'(x) = g'(u(x)) \cdot u'(x).
\]

### Applying the Chain Rule to \( f(x) = (x - 2)^5 \)

1. **Identify the inner function** and **outer function**:
   - **Inner function**: \( u(x) = x - 2 \)
   - **Outer function**: \( g(u) = u^5 \)

2. **Differentiate the inner function** \( u(x) = x - 2 \) with respect to \( x \):
   \[
   u'(x) = \frac{d}{dx}(x - 2) = 1
   \]
   This means the derivative of \( u(x) = x - 2 \) is \( 1 \), because the derivative of \( x \) is \( 1 \), and the derivative of a constant \(-2\) is \( 0 \).

3. **Differentiate the outer function** \( g(u) = u^5 \) with respect to \( u \):
   \[
   g'(u) = \frac{d}{du}(u^5) = 5u^4
   \]

4. **Apply the chain rule**:
   \[
   f'(x) = g'(u(x)) \cdot u'(x)
   \]
   Substituting the derivatives we found:
   \[
   f'(x) = 5(x - 2)^4 \cdot 1
   \]

5. **Simplify**:
   \[
   f'(x) = 5(x - 2)^4
   \]

### Conclusion

The expression \( u'(x) = 1 \) represents the derivative of the inner function \( u(x) = x - 2 \), not the value of the inner function itself. So, when applying the chain rule, we multiply by the derivative of the inner function, which is indeed \( 1 \).

Thank you for catching that. Would you like to ask more questions or need further details?

### Related Questions:

1. What is the derivative of \( (x^2 - 3x + 1)^4 \) using the chain rule?
2. How do you apply the product rule in differentiation?
3. What is the derivative of \( \sin(x^2 + 3x) \)?
4. How do you find the derivative of a function raised to a fractional power, like \( (x + 1)^{3/2} \)?
5. What happens when differentiating functions that have multiple layers of composition?

### Tip:

When differentiating composite functions, always clearly distinguish between the inner and outer functions to avoid confusion in applying the chain rule.

Explore how the chain rule applies to differentiation using a detailed example of (x - 2)^5. Learn the steps to differentiate composite functions correctly, including identifying inner and outer functions and applying the chain rule effectively. Suitable for high school students studying calculus.

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