I've managed to find the derivative to be $f'(x) = \frac{1+e^{x-3} - x(e^{x-3})}{(1+e^{x-3})^2} = 0$
I'm posting a new example because I've noticed too many mistakes in the way I wrote the question that it's easier to scratch it.
I'm stuck on this part but I'm also unsure whether it's correct to do so:
$1+e^{x-3} - x (e^{x-3}) = 0$