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Logarithmic Function and its Derivative

This is a question from my maths book:

Take Logarithms of both sides and use the log laws to differentiate

(a) y=x^x

for this I did:

log y =x log x

then I do not know how to go on from there. I know that the derivative of log x is 1/x

Take ln of both sides:

$ln(y)=ln(x^{x})$

Property of logs:

$ln(y)=xln(x)$

Differentiate:

$\frac{y'}{y}=ln(x)+1$

Solve for y':

$y'=y(ln(x)+1)$

Remember that $y=x^{x}$ and sub in:

$y'=x^{x}(ln(x)+1)$

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