Questions: Evaluate Dx(xx)

Transcript text: Evaluate $D_{x}\left(x^{x}\right)$

Solution

Step 1: Define the Function

Let $ f(x) = x^x $.

Step 2: Take the Natural Logarithm

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

Step 3: Differentiate Both Sides

Differentiate both sides with respect to $ x $: \[ \frac{d}{dx}(\ln(f(x))) = \frac{d}{dx}(x \ln(x)) \]

Using the chain rule on the left side and the product rule on the right side, we get: \[ \frac{1}{f(x)} \cdot f'(x) = \ln(x) + 1 \]

Step 4: Solve for the Derivative

Multiply both sides by $ f(x) $ to isolate $ f'(x) $: \[ f'(x) = f(x)(\ln(x) + 1) \]

Step 5: Substitute Back the Original Function

Substituting back $ f(x) = x^x $: \[ f'(x) = x^x(\ln(x) + 1) \]

$\boxed{f'(x) = x^x(\ln(x) + 1)}$

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