Question 721031
A mistake is found in this part of your process:

"P=2(800/W)+2W -> P=(1600/W)+2W -> P=1600+2W^2"


So you are looking for derivative of perimeter, P, and you should be using {{{P=1600/W+2W}}}.  Derivative with regard to W should be (1600/W)' + 2.  Should be derivative is {{{-1600/(W)^2-2}}}.  The first term would or could use Quotient Rule.