To find the derivative of the function \( f(x) = x^x \), we can use logarithmic differentiation, as the variable \( x \) appears in both the base and the exponent.

### Steps:

1. **Rewrite the Function Using Logarithms:**

   Take the natural logarithm of both sides:
   \[
   y = x^x
   \]
   \[
   \ln y = \ln(x^x)
   \]
   Simplify using the property \( \ln(a^b) = b \ln a \):
   \[
   \ln y = x \ln x
   \]

2. **Differentiate Both Sides with Respect to \( x \):**

   Applying the derivative to both sides, using the chain rule on the left:
   \[
   \frac{1}{y} \cdot \frac{dy}{dx} = \frac{d}{dx} (x \ln x)
   \]

   Now, differentiate \( x \ln x \) using the product rule:
   \[
   \frac{d}{dx} (x \ln x) = \ln x + 1
   \]

   So:
   \[
   \frac{1}{y} \cdot \frac{dy}{dx} = \ln x + 1
   \]

3. **Solve for \( \frac{dy}{dx} \):**

   Substitute \( y = x^x \) back into the equation:
   \[
   \frac{dy}{dx} = x^x (\ln x + 1)
   \]

### Final Answer:

\[
\frac{d}{dx} (x^x) = x^x (\ln x + 1)
\]

Would you like more details on any of these steps?

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Here are some related questions for further exploration:

1. How would you find the second derivative of \( x^x \)?
2. Can you apply logarithmic differentiation to \( x^{x^x} \)?
3. What happens if \( x \) is a complex number in \( x^x \)?
4. How does this derivative compare to \( a^x \), where \( a \) is a constant?
5. How would you integrate \( x^x \)?

**Tip:** Logarithmic differentiation is especially useful for differentiating functions where variables are both in the base and the exponent!

Explore the step-by-step solution to find the derivative of x^x using logarithmic differentiation and the product rule. This solution involves advanced calculus concepts such as logarithmic differentiation and exponentiation. Perfect for college students studying Calculus I or II.

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