Question:
The derivative of \(x^x\) with respect to \(x\) is
The derivative of \(x^x\) with respect to \(x\) is
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For functions like \(x^x\), use logarithmic differentiation.
Updated On: May 7, 2026
- \(x^x(x+\log x)\)
- \(x^x(x-\log x)\)
- \(x^x(1-\log x)\)
- \(x^x(1+\log x)\)
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The Correct Option is D
Solution and Explanation
Step 1: Let: \[ y=x^x \]
Step 2: Taking logarithm: \[ \log y=x\log x \]
Step 3: Differentiate both sides: \[ \frac{1}{y}\frac{dy}{dx}=\log x+1 \]
Step 4: Therefore: \[ \frac{dy}{dx}=y(1+\log x) \]
Step 5: Since \(y=x^x\), \[ \frac{dy}{dx}=x^x(1+\log x) \] \[ \boxed{x^x(1+\log x)} \]
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