Profile likelihood

I am considering a normal distribution with mean $beta_1 + beta_2exp(-phi x)$ and variance $sigma^2$, i.e. $y sim N(beta_1 + beta_2exp(-phi x), sigma^2) $.

My aim is to calculate the profile log likelihood $L_ast(sigma)$ for $sigma$.

I have calculated the log likelihood to be of the form:

$-frac{n}{2}log(2 pi) – n log(sigma) – frac{1}{2 sigma^2} sumlimits_{i=1}^n {(y_i – beta_1 – beta_2exp(-phi x))^2}$

where $n$ is the number of data points

I am trying to show that $L_ast(sigma)$ is of the form $-n log(sigma) – frac{n hat{sigma}^2}{2 sigma^2}$ and then find an expression for $hat{sigma}^2$.

I know that the idea of profile likelihood here is to fix $sigma$ and maximise with respect to the other parameters, i.e. $beta_1, beta_2$ and $phi$. However I have been suggested not to do this by differentiating the likelihood function with respect to these parameters.

I was wondering if anyone had ideas of the best way to go about doing this?