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.