The derivative of `x^(x)` w.r.t. x is ___________.

Answer

To find the derivative of \( x^x \) with respect to \( x \), we can follow these steps:

### Step 1: Rewrite the function
Let \( y = x^x \).

### Step 2: Take the natural logarithm of both sides
Taking the natural logarithm helps us simplify the differentiation:
\[ \ln y = \ln(x^x) \]

### Step 3: Use the properties of logarithms
Using the property of logarithms that states \( \ln(a^b) = b \ln a \):
\[ \ln y = x \ln x \]

### Step 4: Differentiate both sides with respect to \( x \)
Now, we differentiate both sides using implicit differentiation:
\[ \frac{d}{dx}(\ln y) = \frac{d}{dx}(x \ln x) \]

Using the chain rule on the left side:
\[ \frac{1}{y} \frac{dy}{dx} = \frac{d}{dx}(x \ln x) \]

### Step 5: Differentiate the right side
Now we differentiate the right side using the product rule:
\[ \frac{d}{dx}(x \ln x) = \ln x + x \cdot \frac{1}{x} = \ln x + 1 \]

### Step 6: Substitute back into the equation
Now we have:
\[ \frac{1}{y} \frac{dy}{dx} = \ln x + 1 \]

### Step 7: Solve for \(\frac{dy}{dx}\)
Multiplying both sides by \( y \) gives:
\[ \frac{dy}{dx} = y(\ln x + 1) \]

### Step 8: Substitute \( y \) back in terms of \( x \)
Since \( y = x^x \):
\[ \frac{dy}{dx} = x^x (\ln x + 1) \]

### Final Answer
Thus, the derivative of \( x^x \) with respect to \( x \) is:
\[ \frac{dy}{dx} = x^x (\ln x + 1) \]

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