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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