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.

Learn how to differentiate the function (x - 2)^5 with a detailed step-by-step solution. This problem covers differentiation using the power rule and chain rule in calculus. Suitable for advanced high school students and college-level learners.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation