Find the derivative of y = (x^(1/2))^x

Can I get detailed steps on how to complete this? Thanks :D!

lny = x ln (√x)

lny = (x/2) lnx

Differentiate both sides, product rule on right:

y' / y = (x/2)(1/x) + (1/2)(lnx)

y' / y = (1/2) + (1/2)(lnx)

y' / y= (1/2) (1 + lnx)

y' = (!/2)(1+lnx) * y

y' = (1/2)(1+lnx) * (√x)^x

EDIT: probably couldve consildated a little bit of work by knowing that exponents multiply. So it would be ^(x/2) then use ln, i did it the long way, either way would work

Use Logarithmic Differentiation:

ln(y) = xln(x^(1/2)) since ln(a^b) = bln(a)

y'/y = ln*x^(1/2)) + x/(x^(1/2)) * (1/2)x^(-1/2))

y'/y = ln*x^(1/2)) + 1/2

y' = y(ln*x^(1/2)) + 1/2)

y' = (x^(1/2))^x * (ln*x^(1/2)) + 1/2)