Questions: Evaluate Dx(xx)

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

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Solution Steps

Step 1: Define the Function

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

Step 2: Take the Natural Logarithm

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

Step 3: Differentiate Both Sides

Differentiate both sides with respect to x x : ddx(ln(f(x)))=ddx(xln(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: 1f(x)f(x)=ln(x)+1 \frac{1}{f(x)} \cdot f'(x) = \ln(x) + 1

Step 4: Solve for the Derivative

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

Step 5: Substitute Back the Original Function

Substituting back f(x)=xx f(x) = x^x : f(x)=xx(ln(x)+1) f'(x) = x^x(\ln(x) + 1)

Final Answer

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

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