To differentiate the function $f(x) = (x - 2)^5$, we can use the power rule in combination with the chain rule.

Steps to Differentiate $f(x) = (x - 2)^5$:

1. Identify the outer function and the inner function:

   • Outer function: $g(u) = u^5$
   • Inner function: $u(x) = x - 2$

2. Differentiate the outer function $g(u) = u^5$ with respect to $u$: $g'(u) = 5u^4$

3. Differentiate the inner function $u(x) = x - 2$ with respect to $x$: $u'(x) = 1$

4. Apply the chain rule: The chain rule states that if a function $f(x)$ is composed of two functions $g(u(x))$, then $f'(x) = g'(u(x)) \cdot u'(x)$.

   Applying this, we get: $f'(x) = g'(u(x)) \cdot u'(x)$ Substituting in the derivatives we found: $f'(x) = 5(x - 2)^4 \cdot 1$

5. Simplify the expression: $f'(x) = 5(x - 2)^4$

Final Answer:

$\frac{d}{dx}[(x - 2)^5] = 5(x - 2)^4$

Would you like a detailed explanation or have any questions about this differentiation?

Related Questions:

1. How do you apply the power rule to differentiate other polynomial functions?
2. What is the chain rule, and how is it used in calculus?
3. How would you differentiate $(x^2 - 3x + 1)^6$?
4. What is the general formula for the derivative of $(ax + b)^n$?
5. How do you find the derivative of a product of functions?

Tip:

When differentiating composite functions, remember to identify the outer and inner functions first, and then apply the chain rule carefully.