https://file.aiquickdraw.com/mathbot/file/1728690475840n5j5v810.jpg

Let's solve the given derivative problems step by step by matching the function \( f(x) \) with its corresponding derivative \( f'(x) \).

### Functions:
- \( f(x) = \frac{2}{x} \)
- \( f(x) = \ln(x) \)
- \( f(x) = \ln(x^2) \)
- \( f(x) = x \)
- \( f(x) = \frac{1}{x} \)
- \( f(x) = e^x \)

### Derivatives:
1. \( f'(x) = \frac{-2}{x^2} \)
2. \( f'(x) = \frac{2}{x} \)
3. \( f'(x) = \frac{-1}{x^2} \)
4. \( f'(x) = \frac{1}{x} \)
5. \( f'(x) = 1 \)
6. \( f'(x) = e^x \)

Now, let's find the derivatives for each of the provided functions and match them:

1. **\( f(x) = \frac{2}{x} \)**  
   Derivative:  
   \[
   f'(x) = \frac{d}{dx} \left( \frac{2}{x} \right) = \frac{-2}{x^2}
   \]  
   This matches derivative \( f'(x) = \frac{-2}{x^2} \).

2. **\( f(x) = \ln(x) \)**  
   Derivative:  
   \[
   f'(x) = \frac{d}{dx} \left( \ln(x) \right) = \frac{1}{x}
   \]  
   This matches derivative \( f'(x) = \frac{1}{x} \).

3. **\( f(x) = \ln(x^2) \)**  
   Using the logarithmic property \( \ln(x^2) = 2 \ln(x) \), its derivative is:  
   \[
   f'(x) = \frac{d}{dx} \left( 2 \ln(x) \right) = \frac{2}{x}
   \]  
   This matches derivative \( f'(x) = \frac{2}{x} \).

4. **\( f(x) = x \)**  
   Derivative:  
   \[
   f'(x) = \frac{d}{dx} (x) = 1
   \]  
   This matches derivative \( f'(x) = 1 \).

5. **\( f(x) = \frac{1}{x} \)**  
   Derivative:  
   \[
   f'(x) = \frac{d}{dx} \left( \frac{1}{x} \right) = \frac{-1}{x^2}
   \]  
   This matches derivative \( f'(x) = \frac{-1}{x^2} \).

6. **\( f(x) = e^x \)**  
   Derivative:  
   \[
   f'(x) = \frac{d}{dx} \left( e^x \right) = e^x
   \]  
   This matches derivative \( f'(x) = e^x \).

### Correct Matches:
1. \( f(x) = \frac{2}{x} \) matches \( f'(x) = \frac{-2}{x^2} \)
2. \( f(x) = \ln(x^2) \) matches \( f'(x) = \frac{2}{x} \)
3. \( f(x) = \frac{1}{x} \) matches \( f'(x) = \frac{-1}{x^2} \)
4. \( f(x) = \ln(x) \) matches \( f'(x) = \frac{1}{x} \)
5. \( f(x) = x \) matches \( f'(x) = 1 \)
6. \( f(x) = e^x \) matches \( f'(x) = e^x \)

Would you like more details on any specific part of this solution?

### Related Questions:
1. How do you apply the chain rule in differentiation?
2. What is the derivative of a natural logarithmic function?
3. How does the power rule apply to \( f(x) = x^n \)?
4. Can you explain logarithmic differentiation with examples?
5. What is the significance of the derivative of exponential functions?

### Tip:
When differentiating logarithmic functions, use properties like \( \ln(a^b) = b \ln(a) \) to simplify before finding the derivative.

Match each function f(x) with its corresponding derivative f'(x) from the given options.

Learn how to match functions like ln(x), e^x, and 1/x with their corresponding derivatives. This exercise covers key concepts in logarithmic and exponential differentiation, making it ideal for advanced high school and early university calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation