To differentiate \( y = x^x \), we'll proceed as follows:

### Step 1: Rewrite the function using logarithms
Given:
\[
y = x^x
\]

Take the natural logarithm on both sides:
\[
\ln(y) = \ln(x^x)
\]

Using the logarithmic identity \( \ln(a^b) = b \ln(a) \), we get:
\[
\ln(y) = x \ln(x)
\]

### Step 2: Differentiate implicitly
Differentiate both sides with respect to \( x \):
\[
\frac{1}{y} \frac{dy}{dx} = \frac{d}{dx}(x \ln(x))
\]

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

Therefore:
\[
\frac{1}{y} \frac{dy}{dx} = \ln(x) + 1
\]

### Step 3: Solve for \( \frac{dy}{dx} \)
Multiply both sides by \( y \):
\[
\frac{dy}{dx} = y(\ln(x) + 1)
\]

Substituting \( y = x^x \):
\[
\frac{dy}{dx} = x^x (\ln(x) + 1)
\]

### Final answer
\[
\frac{dy}{dx} = x^x (\ln(x) + 1)
\]

Would you like more details or have any questions?

### Related Questions
1. How do you differentiate \( y = x^{x^2} \)?
2. How would you integrate \( x^x \)?
3. What is the derivative of \( y = x^a \) where \( a \) is a constant?
4. How do you find the second derivative of \( y = x^x \)?
5. How does the differentiation change for \( y = a^x \) where \( a \) is a constant?

### Tip
Always consider using logarithmic differentiation when both the base and exponent involve the variable, as it simplifies the process.

Learn how to differentiate y = x^x using logarithmic differentiation. This step-by-step guide covers key calculus concepts such as the product rule and implicit differentiation, making it suitable for students in Grades 11-12 and early college calculus courses.

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