Let X1...Xn be a random sample from a Poisson distribution with mean \mu. Let Y = \Sigma X_i has a Poisson distribution with mean n \mu. \overline{X} = Y/n is approximately N(\mu, \mu/n for large n. Show that u(Y/n) = \sqrt{Y/n} is a function of Y/n whose variance is free of \mu

u(Y/n) = u(\overline{X}) = u(\mu) + u(\mu)(\overline{X})
Var[u(\overline{X})] = [u'(\mu]^2(\mu/n) = c
u'(\mu) = c_1/\sqrt{\mu} (c1 = cn)
u(\mu) = c2 \sqrt{\mu} (c2 = 2c1)

If c2 = 1, then u(\overline{X}) = \sqrt{X} = \sqrt{Y/n}

Is this correct, or have I made a mistake somewhere?