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Worked solution

Prove the derivative of x to the power x

Prove by differentiation that d(xx)/dx=xx(1+ln⁡x)d(x^x)/dx=x^x(1+\ln x).

CJC · Lecture · Self-Practice 5

Take natural logarithms on the positive domain

For x>0x>0, let y=xx>0y=x^x>0. Then ln⁡y=xln⁡x\ln y=x\ln x. Differentiating gives y′/y=ln⁡x+1y^{\prime}/y=\ln x+1.

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Restore the original function

Multiply by y=xxy=x^x to obtain y′=xx(1+ln⁡x)y^{\prime}=x^x(1+\ln x), the requested printed result. Both the base and the exponent vary, so treating either one as constant would omit a term.

How did this part go?

Common mistakes

The power rule with a constant exponent does not apply directly to x^x.

Solution notes

Both originals print the asks and final answers but no intermediate working. The intermediate derivations and domain explanations here are independently authored and checked against the printed answers. No printed marks or past-paper attribution.

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