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Out of genuine curiosity, does d/dx of x^x=x^x?

Mathematics
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My logic is that d/dx(x^x)= (x)(x^(x-1))= x^((x-1)+1)= x^x
\[y=x^x \] Use log diff: First step take log of both sides: \[\ln(y)=\ln(x^x) => \ln(y) =x \ln(x) \] Now differentiate both sides: Then solve for y'
Anyways, I can finish the problem if you want

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I don't want to waste any more of your time. You answered my curiosity, and that's all I needed. Thank you very much.
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