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.